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The multivariate time series models used in Econometrics Toolbox™ functions are based on linear, autoregressive models. The basic models are:
Model Name | Abbreviation | Equation |
Vector Autoregressive | VAR(p) | |
Vector Moving Average | VMA(q) | |
Vector Autoregressive Moving Average | VARMA(p, q) | |
Vector Autoregressive Moving Average with eXogenous inputs | VARMAX(p, q, r) | |
Structural Vector Autoregressive Moving Average with eXogenous inputs | SVARMAX(p, q, r) | |
The following variables appear in the equations:
yt is the vector of response time series variables at time t. yt has n elements.
a is a constant vector of offsets, with n elements.
Ai are n-by-n matrices for each i. The Ai are autoregressive matrices. There are p autoregressive matrices.
εt is a vector of serially uncorrelated innovations, vectors of length n. The εt are multivariate normal random vectors with a covariance matrix Q, where Q is an identity matrix, unless otherwise specified.
Bj are n-by-n matrices for each j. The Bj are moving average matrices. There are q moving average matrices.
Xt is an n-by-r matrix representing exogenous terms at each time t. r is the number of exogenous series. Exogenous terms are data (or other unmodeled inputs) in addition to the response time series yt.
b is a constant vector of regression coefficients of size r. So the product Xt·b is a vector of size n.
Generally, the time series yt and Xt are observable. In other words, if you have data, it represents one or both of these series. You do not always know the offset a, coefficient b, autoregressive matrices Ai, and moving average matrices Bj. You typically want to fit these parameters to your data. See the vgxvarx function reference page for ways to estimate unknown parameters. The innovations εt are not observable, at least in data, though they can be observable in simulations.
There is an equivalent representation of the linear autoregressive equations in terms of lag operators. The lag operator L moves the time index back by one: Lyt = yt–1. The operator Lm moves the time index back by m: Lmyt = yt–m.
In lag operator form, the equation for a SVARMAX(p, q, r) model becomes
This equation can be written as
This condition implies that, with all innovations equal to zero, the VAR process converges to a as time goes on. See Lütkepohl [69] Chapter 2 for a discussion.
This condition implies that the pure VAR representation of the process is stable. For an explanation of how to convert between VAR and VMA models, see Changing Model Representations. See Lütkepohl [69] Chapter 11 for a discussion of invertible VMA models.
A VARMA model is stable if its VAR part is stable. Similarly, a VARMA model is invertible if its VMA part is invertible.
There is no well-defined notion of stability or invertibility for models with exogenous inputs (e.g., VARMAX models). An exogenous input can destabilize a model.
To understand a multiple time series model, or multiple time series data, you generally perform the following steps:
Import and preprocess data.
Specify a model.
Specifying Models to set up a model using vgxset:
Specification Structures with Known Parameters to specify a model with known parameters
Specification Structures with No Parameter Values to specify a model when you want MATLAB® to estimate the parameters
Specification Structures with Selected Parameter Values to specify a model where you know some parameters, and want MATLAB to estimate the others
Determining an Appropriate Number of Lags to determine an appropriate number of lags for your model
Fit the model to data.
Fitting Models to Data to use vgxvarx to estimate the unknown parameters in your models. This can involve:
Changing Model Representations to change your model to a type that vgxvarx handles
Estimating Structural Matrices
Analyze and forecast using the fitted model. This can involve:
Checking Stability to determine whether your model is stable and invertible
Forecasting with vgxpred to forecast directly from models
Forecasting with vgxsim to simulate a model
Example: Calculating Impulse Responses with vgxproc to calculate impulse responses, which give forecasts based on an assumed change in an input to a time series
Comparing Forecasts with Forecast Period Data to compare the results of your model's forecasts to your data
Your application need not involve all of the steps in this workflow. For example, you might not have any data, but want to simulate a parameterized model. In that case, you would perform only steps 2 and 4 of the generic workflow.
You might iterate through some of these steps.
Often, the first step in creating a multiple time series model is to obtain data. There are two types of multiple time series data:
Response data. Response data corresponds to yt in the multiple time series models defined in Types of VAR Models.
Exogenous data. Exogenous data corresponds to Xt in the multiple time series models defined in Types of VAR Models.
Before using Econometrics Toolbox functions with the data, put the data into the required form. Use standard MATLAB commands, or preprocess the data with a spreadsheet program, database program, PERL, or other utilities.
There are several freely available sources of data sets, such as the St. Louis Federal Reserve Economics Database (known as FRED): http://research.stlouisfed.org/fred2/. If you have a license, you can use Datafeed Toolbox™ functions to access data from various sources.
Response data for multiple time series models must be in the form of a matrix. Each row of the matrix represents one time, and each column of the matrix represents one time series. The earliest data is the first row, the latest data is the last row. The data represents yt in the notation of Types of VAR Models. If there are T times and n time series, put the data in the form of a T-by-n matrix:
Y1t represents time series 1,..., Ynt represents time series n, 1 ≤ t ≤ T.
Multiple Paths. Response time series data can have an extra dimension corresponding to separate, independent paths. For this type of data, use a three-dimensional array Y(t,j,p), where:
t is the time index of an observation, 1 ≤ t ≤ T.
j is the index of a time series, 1 ≤ j ≤ n.
p is the path index, 1 ≤ p ≤ P.
For any path p, Y(t,j,p) is a time series.
The file Data_USEconModel ships with Econometrics Toolbox software. It contains time series from the St. Louis Federal Reserve Economics Database (known as FRED).
load Data_USEconModel
to load the data into your MATLAB workspace. The following items load into the workspace:
Data, a 250-by-14 matrix containing the time series of data,
Dataset, a 250-by-14 matrix containing the dataset array,
Description, a character array containing a description of the data series and the key to the labels for each series,
dates, a 250-element vector containing the dates for Data,
series, a 1-by-14 cell array of labels for the time series.
Examine the data structures:
load Data_USEconModel dates(1) ans = = 711217 dates(end) ans = = 733954
dates are MATLAB serial date numbers, the number of days since the putative date January 1, 0000. (This "date" is not a real date, but is convenient for making date calculations; for more information, see Date Formats in the Financial Toolbox™ User's Guide.)
The Data matrix contains 14 columns. These represent the time series labeled by Series.
FRED Series | Description |
COE | Paid compensation of employees in $ billions |
CPIAUCSL | Consumer Price Index |
FEDFUNDS | Effective federal funds rate |
GCE | Government consumption expenditures and investment in $ billions |
GDP | Gross Domestic Product |
GDPDEF | Gross domestic product in $ billions |
GDPI | Gross private domestic investment in $ billions |
GS10 | Ten-year treasury bond yield |
HOANBS | Non-farm business sector index of hours worked |
M1SL | M1 money supply (narrow money) |
M2SL | M2 money supply (broad money) |
PCEC | Personal consumption expenditures in $ billions |
TB3MS | Three-month treasury bill yield |
UNRATE | Unemployment rate |
The data for exogenous inputs, Xt, has a different form. Put the data in a T-by-1 matrix of cell arrays (a column vector of cell arrays). Each cell array is an n-by-r matrix, where n is the number of time series in the response data structure, and r is the number of exogenous time series. The n-by-r matrix consists of the usual MATLAB floating-point numbers.
At each time t, the n-by-r matrix Xt multiplies the r-by-1 vector b, yielding an n-vector.
This structure is augmented in the case of multiple paths. If there are p paths, put the data in a T-by-p matrix of cell arrays. Each entry in the cell array again is an n-by-r matrix.
Example: Exogenous Data Structure. This example uses linear exogenous data. Suppose you have an T-by-3 data series. The exogenous series is a T-element cell array of 3-by-3 matrices, where cell i is i*eye(3):
xdata = cell(T,1); for indx = 1:T xdata{indx} = indx * eye(3); end xdata{1} ans = 1 0 0 0 1 0 0 0 1 xdata{T} % In this example, T = 10 ans = 10 0 0 0 10 0 0 0 10
Your data might have characteristics that violate assumptions for linear multiple time series models. For example, you can have data with exponential growth, or data from multiple sources at different periodicities. You must preprocess your data to convert it into an acceptable form for analysis.
For time series with exponential growth, you can preprocess the data by taking the logarithm of the growing series. In some cases you then difference the result. For an example, see Transforming Data for Stationarity.
For data from multiple sources, you must decide how best to fill in missing values. Commonly, you take the missing values as unchanged from the previous value, or as interpolated from neighboring values.
Note: If you take a difference of a series, the series becomes 1 shorter. If you take a difference of only some time series in a data set, truncate the other series so that all have the same length, or pad the differenced series with initial values. |
Testing Data for Stationarity. You can test each time series data column for stability using unit root tests. For details, see Unit Root Nonstationarity.
To fit a lagged model to data, partition the response data in up to three sections:
Presample data
Estimation data
Forecast data
The purpose of presample data is to provide initial values for lagged variables. When trying to fit a model to the estimation data, you need to access earlier times. For example, if the maximum lag in a model is 4, and if the earliest time in the estimation data is 50, then the model needs to access data at time 46 when fitting the observations at time 50. vgxvarx assumes the value 0 for any data that is not part of the presample data.
The estimation data contains the observations yt. Use the estimation data to fit the model.
Use the forecast data for comparing fitted model predictions against data. You do not have to have a forecast period. Use one to validate the predictive power of a fitted model.
The following figure shows how to arrange the data in the data matrix, with j presample rows and k forecast rows.
The following discussion refers to one path. The ideas apply to multiple paths as well.
Exogenous variables affect the response time series through multiplication by the regression coefficient b. For more information, see Types of VAR Models. At each time index, the exogenous data is an n-by-r matrix, where n is the number of time series in the multiple time series model, and r is the number of exogenous time series.
To give the vgxvarx fitting function more flexibility in fitting the effect of the exogenous series on each response time series, expand or replicate your exogenous time series. This is the technique of seemingly unrelated regression (SUR).
Suppose, for example, a tariff is imposed over some time period. You suspect that this tariff affects GNP and other economic quantities. Suppose you are examining data that contains four time series. Here are two ways of including the exogenous tariffs:
Each cell in the exogenous time series consists of either
ones(4,1) or zeros(4,1):
a 4–by-1 matrix indicating the presence (1) or absence (0) of the tariff during the time in question.
Each cell in the exogenous time series consists of either
eye(4) or zeros(4):
a 4–by-4 matrix indicating the presence (1) or absence (0) of the tariff during the time in question.
The advantage of the larger (replicated) formulation is that it allows for vgxvarx to estimate the influence of the tariff on each time series separately. The resulting regression coefficient vector b can have differing values for each component. The different values reflect the different direct influences of the tariff on each time series.
Similarly, if you have an exogenous series X(t), an n-by-1 vector for each time t, you can include it in your formulation as diag(X), an n-by-n matrix.
You can model linear trends in your data by including the exogenous matrix eye(n)*t at time index t.
Econometrics Toolbox functions require a model specification structure as an input before they simulate, calibrate, forecast, or perform other calculations. You can specify a model with or without time series data. If you have data, you can fit models to the data as described in VAR Model Estimation. If you do not have data, you can specify a model with parameters you provide, as described in Specification Structures with Known Parameters.
Create an Econometrics Toolbox multiple time series model specification structure using the vgxset function. Use this structure for calibrating, simulating, predicting, analyzing, and displaying multiple time series.
There are three ways to create model structures using the vgxset function:
Specification Structures with Known Parameters. Use this method when you know the values of all relevant parameters of your model.
Specification Structures with No Parameter Values. Use this method when you know the size, type, and number of lags in your model, but do not know the values of any of the AR or MA coefficients, or the value of your Q matrix.
Specification Structures with Selected Parameter Values. Use this method when you know the size, type, and number of lags in your model, and also know some, but not all, of the values of AR or MA coefficients. This method includes the case when you want certain parameters to be zero. You can specify as many parameters as you like. For example, you can specify certain parameters, request that vgxvarx estimate others, and specify other parameters with [] to indicate default values.
If you know the values of model parameters, create a model structure with the parameters. The following are the name-value pairs you can pass to vgxset for known parameter values:
Model Parameters
Name | Value |
a | An n-vector of offset constants. The default is empty. |
AR0 | An n-by-n invertible matrix representing the zero-lag structural AR coefficients. The default is empty, which means an identity matrix. |
AR | An nAR-element cell array of n-by-n matrices of AR coefficients. The default is empty. |
MA0 | An n-by-n invertible matrix representing the zero-lag structural MA coefficients. The default is empty, which means an identity matrix. |
MA | An nMA-element cell array of n-by-n matrices of MA coefficients. The default is empty. |
b | An nX-vector of regression coefficients. The default is empty. |
Q | An n-by-n symmetric innovations covariance matrix. The default is empty, which means an identity matrix. |
ARlag | A monotonically increasing nAR-vector of AR lags. The default is empty, which means all the lags from 1 to p, the maximum AR lag. |
MAlag | A monotonically increasing nMA-vector of MA lags. The default is empty, which means all the lags from 1 to q, the maximum MA lag. |
vgxset infers the model dimensionality, given by n, p, and q in Types of VAR Models, from the input parameters. These parameters are n, nAR, and nMA respectively in vgxset syntax. For more information, see Specification Structures with No Parameter Values.
The ARlag and MAlag vectors allow you to specify which lags you want to include. For example, to specify AR lags 1 and 3 without lag 2, set ARlag to [1 3]. This setting corresponds to nAR = 2 for two specified lags, even though this is a third order model, since the maximum lag is 3.
The following example shows how to create a model structure when you have known parameters. Consider a VAR(1) model:
Specifically, a = [0.05, 0, –.05]' and wt are distributed as standard three-dimensional normal random variables.
Create a model specification structure with vgxset:
A1 = [.5 0 0;.1 .1 .3;0 .2 .3]; Q = eye(3); armodel = vgxset('a',[.05;0;-.05],'AR',{A1},'Q',Q) armodel = Model: 3-D VAR(1) with Additive Constant n: 3 nAR: 1 nMA: 0 nX: 0 a: [0.05 0 -0.05] additive constants AR: {1x1 cell} stable autoregressive process Q: [3x3] covariance matrix
vgxset identifies this model as a stable VAR(1) model with three dimensions and additive constants.
By default, vgxvarx fits all unspecified additive (a), AR, regression coefficients (b), and Q parameters. You must specify the number of time series and the type of model you want vgxvarx to fit. The following are the name-value pairs you can pass to vgxset for unknown parameter values:
Model Orders
Name | Value |
n | A positive integer specifying the number of time series. The default is 1. |
nAR | A nonnegative integer specifying the number of AR lags (corresponds to p in Types of VAR Models). The default is 0. |
nMA | A nonnegative integer specifying the number of MA lags (corresponds to q in Types of VAR Models). The default is 0. Currently, vgxvarx cannot fit MA matrices. Therefore, specifying an nMA greater than 0 does not yield estimated MA matrices. |
nX | A nonnegative integer specifying the number regression parameters (corresponds to r in Types of VAR Models). The default is 0. |
Constant | Additive offset logical indicator. The default is false. |
The following example shows how to specify the model in Specification Structures with Known Parameters, but without explicit parameters.
arunknown = vgxset('n',3,'nAR',1,'Constant',true) arunknown = Model: 3-D VAR(1) with Additive Constant n: 3 nAR: 1 nMA: 0 nX: 0 a: [] AR: {} Q: []
You can create a model structure with some known parameters, and have vgxvarx fit the unknown parameters to data. Here are the name-value pairs you can pass to vgxset for requested parameter values:
Model Parameter Estimation
Name | Value |
asolve | An n-vector of additive offset logical indicators. The default is empty, which means true(n,1). |
ARsolve | An nAR-element cell array of n-by-n matrices of AR logical indicators. The default is empty, which means an nAR-element cell array of true(n). |
AR0solve | An n-by-n matrix of AR0 logical indicators. The default is empty, which means false(n). |
MAsolve | An nMA-element cell array of n-by-n matrices of MA logical indicators. The default is empty, which means false(n). |
MA0solve | An n-by-n matrix of MA0 logical indicators. The default is empty, which means false(n). |
bsolve | An nX-vector of regression logical indicators. The default is empty, which means true(n,1). |
Qsolve | An n-by-n symmetric covariance matrix logical indicator. The default is empty, which means true(n), unless the 'CovarType' option of vgxvarx overrides it. |
Specify a logical 1 (true) for every parameter that you want vgxvarx to estimate.
Currently, vgxvarx cannot fit the AR0, MA0, or MA matrices. Therefore, vgxvarx ignores the AR0solve, MA0solve, and MAsolve indicators. However, you can examine the Example_StructuralParams.m file for an approach to estimating the AR0 and MA0 matrices. Enter help Example_StructuralParams at the MATLAB command line for information. See Lütkepohl [69] Chapter 9 for algorithms for estimating structural models.
Currently, vgxvarx also ignores the Qsolve matrix. vgxvarx can fit either a diagonal or a full Q matrix; see vgxvarx.
This example shows how to specify the model in Specification Structures with Known Parameters, but with requested AR parameters with a diagonal autoregressive structure. The dimensionality of the model is known, as is the parameter vector a, but the autoregressive matrix A1 and covariance matrix Q are not known.
arrequested = vgxset('ARsolve',{logical(eye(3))},'a',... [.05;0;-.05]) arrequested = Model: 3-D VAR(1) with Additive Constant n: 3 nAR: 1 nMA: 0 nX: 0 a: [0.05 0 -0.05] additive constants AR: {} ARsolve: {1x1 cell of logicals} autoregressive lag indicators Q: []
After you set up a model structure, you can examine it in several ways:
Call the vgxdisp function.
Double-click the structure in the MATLAB Workspace browser.
Call the vgxget function.
Enter Spec at the MATLAB command line, where Spec is the name of the model structure.
Enter Spec.ParamName at the MATLAB command line, where Spec is the name of the model structure, and ParamName is the name of the parameter you want to examine.
You can change any part of a model structure named, for example, Spec, using the vgxset as follows:
Spec = vgxset(Spec,'ParamName',value,...)
This syntax changes only the 'ParamName' parts of the Spec structure.
There are two Econometrics Toolbox functions that can help you determine an appropriate number of lags for your models:
The lratiotest function performs likelihood ratio tests to help identify the appropriate number of lags.
The aicbic function calculates the Akaike and Bayesian information criteria to determine the minimal appropriate number of required lags.
Example: Using the Likelihood Ratio Test to Calculate the Minimal Requisite Lag. lratiotest requires inputs of the loglikelihood of an unrestricted model, the loglikelihood of a restricted model, and the number of degrees of freedom (DoF). DoF is the difference in the number of active parameters between the unrestricted and restricted models. lratiotest returns a Boolean: 1 means reject the restricted model in favor of the unrestricted model, 0 means there is insufficient evidence to reject the restricted model.
In the context of determining an appropriate number of lags, the restricted model has fewer lags, and the unrestricted model has more lags. Otherwise, test models with the same type of fitted parameters (for example, both with full Q matrices, or both with diagonal Q matrices).
Obtain the loglikelihood (LLF) of a model as the third output of vgxvarx:
[EstSpec,EstStdErrors,LLF,W] = vgxvarx(...)
Obtain the number of active parameters in a model as the second output of vgxcount:
[NumParam,NumActive] = vgxcount(Spec)
For example, suppose you have four fitted models with varying lag structures. The models have loglikelihoods LLF1, LLF2, LLF3, and LLF4, and active parameter counts n1p, n2p, n3p, and n4p. Suppose model 4 has the largest number of lags. Perform likelihood ratio tests of models 1, 2, and 3 against model 4, as follows:
reject1 = lratiotest(LLF4,LLF1,n4p - n1p) reject2 = lratiotest(LLF4,LLF2,n4p - n2p) reject3 = lratiotest(LLF4,LLF3,n4p - n3p)
If reject1 = 1, you reject model 1 in favor of model 4, and similarly for models 2 and 3. If any of the models have rejectI = 0, you have an indication that you can use fewer lags than in model 4.
Example: Using Akaike Information Criterion to Calculate the Minimal Requisite Lag. aicbic requires inputs of the loglikelihood of a model and of the number of active parameters in the model. It returns the value of the Akaike information criterion. Lower values are better than higher values. aicbic accepts vectors of loglikelihoods and vectors of active parameters, and returns a vector of Akaike information criteria, which makes it easy to find the minimum.
Obtain the loglikelihood (LLF) of a model as the third output of vgxvarx:
[EstSpec,EstStdErrors,LLF,W] = vgxvarx(...)
Obtain the number of active parameters in a model as the second output of vgxcount:
[NumParam,NumActive] = vgxcount(Spec)
For example, suppose you have four fitted models with varying lag structures. The models have loglikelihoods LLF1, LLF2, LLF3, and LLF4, and active parameter counts n1p, n2p, n3p, and n4p. Calculate the Akaike information criteria as follows:
AIC = aicbic([LLF1 LLF2 LLF3 LLF4],[n1p n2p n3p n4p])
The most suitable model has the lowest value of the Akaike information criterion.
To create a model of multiple time series data, decide on a parametric form of the model, and fit parameters to the data. When you have a calibrated (fitted) model, check if the model fits the data adequately.
To fit a model to data, you must have:
Time series data, as described in Multivariate Time Series Data
At least one time series model specification structure, as described in Model Specification Structures
There are several Econometrics Toolbox functions that aid these tasks, including:
vgxvarx, which fits VARX models.
vgxar and vgxma, which convert models to pure AR or MA models; vgxar enables you to fit VARMA models with vgxvarx, as described in Example: Fitting a VARMA Model
lratiotest, lmtest, waldtest, and aicbic, which can help determine the number of lags to include in a model.
vgxqual, which examines the stability of models, as described in Checking Model Adequacy.
vgxpred, which creates forecasts that can be used to check the adequacy of the fit, as described in VAR Model Forecasting, Simulation, and Analysis
Structural Matrices. The structural matrices in SVARMAX models are the A0 and B0 matrices. See Types of VAR Models for definitions of these terms. Currently, vgxvarx cannot fit these matrices to data. However, you can examine the Example_StructuralParams.m file for an approach to estimating the AR0 and MA0 matrices. Enter help Example_StructuralParams at the MATLAB command line for information. See Lütkepohl [69] Chapter 9 for algorithms for estimating structural models.
Include exogenous data in a VARMAX model Xt by setting the nX field to the number of exogenous time series. See Types of VAR Models for definitions of these terms. To fit the coefficient vector b (which multiplies the exogenous matrices), create your model structure with vgxset to have either bsolve = true(r,1), where r is the number of elements in b, or have nX defined as the number of elements in b. By default, vgxvarx fits all elements of the b vector.
You can convert a VARMA model to an equivalent VAR model using the vgxar function. (See Types of VAR Models for definitions of these terms.) Similarly, you can convert a VARMA model to an equivalent VMA model using the vgxma function. These functions are useful in the following situations:
Calibration of models
The vgxvarx function can calibrate only VAR and VARX models. So to calibrate a VARMA model, you could first convert it to a VAR model. However, you can ask vgxvarx to ignore VMA terms and fit just the VAR structure. See Example: Fitting a VARMA Model for a comparison of conversion versus no conversion.
It is straightforward to generate forecasts for VMA models. In fact, vgxpred internally converts models to VMA models to calculate forecast statistics.
Analyzing models
Sometimes it is easier to define your model using one structure, but you want to analyze it using a different structure.
The algorithm for conversion between models involves series that are, in principle, infinite. The vgxar and vgxma functions truncate these series to the maximum of nMA and nAR, introducing an inaccuracy. You can specify that the conversion give more terms, or give terms to a specified accuracy. See [69] for more information on these transformations.
Example: Converting a VARMA Model to a VAR Model. This example creates a VARMA model, then converts it to a pure VAR model.
Create a VARMA model specification structure.
A1 = [.2 -.1 0;.1 .2 .05;0 .1 .3]; A2 = [.3 0 0;.1 .4 .1;0 0 .2]; A3 = [.4 .1 -.1;.2 -.5 0;.05 .05 .2]; MA1 = .2*eye(3); MA2 = [.3 .2 .1;.2 .4 0;.1 0 .5]; Spec = vgxset('AR',{A1,A2,A3},'MA',{MA1,MA2}) Spec = Model: 3-D VARMA(3,2) with No Additive Constant n: 3 nAR: 3 nMA: 2 nX: 0 AR: {3x1 cell} stable autoregressive process MA: {2x1 cell} invertible moving average process Q: []
Convert the structure to a pure VAR structure:
SpecAR = vgxar(Spec) SpecAR = Model: 3-D VAR(3) with No Additive Constant n: 3 nAR: 3 nMA: 0 nX: 0 AR: {3x1 cell} unstable autoregressive process Q: []
The converted process is unstable; see the AR row. An unstable model could yield inaccurate predictions. Taking more terms in the series gives a stable model:
specAR4 = vgxar(Spec,4) specAR4 = Model: 3-D VAR(4) with No Additive Constant n: 3 nAR: 4 nMA: 0 nX: 0 AR: {4x1 cell} stable autoregressive process Q: []
Example: Converting a VARMA Model to a VMA Model. This example uses a VARMA model and converts it to a pure VMA model.
Convert the model specification structure to a pure MA structure:
SpecMA = vgxma(Spec) SpecMA = Model: 3-D VMA(3) with No Additive Constant n: 3 nAR: 0 nMA: 3 nX: 0 MA: {3x1 cell} non-invertible moving average process Q: []
Notice that the pure VMA process has more MA terms than the original process. The number is the maximum of nMA and nAR, and nAR = 3.
The converted VMA model is not invertible; see the MA row. A noninvertible model can yield inaccurate predictions. Taking more terms in the series results in an invertible model.
specMA4 = vgxma(Spec,4) specMA4 = Model: 3-D VMA(4) with No Additive Constant n: 3 nAR: 0 nMA: 4 nX: 0 MA: {4x1 cell} invertible moving average process Q: []
Converting the resulting VMA model to a pure VAR model results in the same model as calculated in Example: Converting a VARMA Model to a VAR Model:
SpecAR2 = vgxar(SpecMA); vgxdisp(SpecAR,SpecAR2) Model : 3-D VAR(3) with No Additive Constant Conditional mean is not AR-stable and is MA-invertible Model : 3-D VAR(3) with No Additive Constant Conditional mean is not AR-stable and is MA-invertible Parameter Model 1 Model 2 -------------- -------------- -------------- AR(1)(1,1) 0.4 0.4 (1,2) -0.1 -0.1 (1,3) -0 -0 (2,1) 0.1 0.1 (2,2) 0.4 0.4 (2,3) 0.05 0.05 (3,1) -0 -0 (3,2) 0.1 0.1 (3,3) 0.5 0.5 AR(2)(1,1) 0.52 0.52 (1,2) 0.22 0.22 (1,3) 0.1 0.1 (2,1) 0.28 0.28 (2,2) 0.72 0.72 (2,3) 0.09 0.09 (3,1) 0.1 0.1 (3,2) -0.02 -0.02 (3,3) 0.6 0.6 AR(3)(1,1) 0.156 0.156 (1,2) -0.004 -0.004 (1,3) -0.18 -0.18 (2,1) 0.024 0.024 (2,2) -0.784 -0.784 (2,3) -0.038 -0.038 (3,1) -0.01 -0.01 (3,2) 0.014 0.014 (3,3) -0.17 -0.17 Q(:,:) [] []
Conversion Types and Accuracy. Some conversions occur when explicitly requested, such as those initiated by calls to vgxar and vgxma. Other conversions occur automatically as needed for calculations. For example, vgxpred internally converts models to VMA models to calculate forecast statistics.
By default, conversions give terms up to the largest lag present in the model. However, for more accuracy in conversion, you can specify that the conversion use more terms. You can also specify that it continue until a residual term is below a threshold you set. The syntax is
SpecAR = vgxar(Spec,nAR,ARlag,Cutoff) SpecMA = vgxma(Spec,nMA,MAlag,Cutoff)
nMA and nAR represent the number of terms in the series.
ARlag and MAlag are vectors of particular lags that you want in the converted model.
Cutoff is a positive parameter that truncates the series if the norm of a converted term is smaller than Cutoff. Cutoff is 0 by default.
For details, see the function reference pages for vgxar and vgxma.
The vgxvarx function performs parameter estimation. vgxvarx only estimates parameters for VAR and VARX models. In other words, vgxvarx does not estimate moving average matrices, which appear, for example, in VMA and VARMA models. For definitions of these terms, see Types of VAR Models.
The vgxar function converts a VARMA model to a VAR model. Currently, it does not handle VARMAX models.
You have two choices in fitting parameters to a VARMA model or VARMAX model:
Set the vgxvarx 'IgnoreMA' parameter to 'yes' (the default is 'no'). In this case vgxvarx ignores VMA parameters, and fits the VARX parameters.
Convert a VARMA model to a VAR model using vgxar. Then fit the resulting VAR model using vgxvarx.
Each of these options is effective on some data. Try both if you have VMA terms in your model. See Example: Fitting a VARMA Model for an example showing both options.
Example: Fitting a VAR Model. This example uses two series: the consumer price index (CPI) and the unemployment rate (UNRATE) from the data set Data_USEconmodel.
Obtain the two time series, and convert them for stationarity:
load Data_USEconModel CPI = Dataset.CPIAUCSL; CPI = log(CPI); CPID = diff(CPI); DatesD = dates(2:end); UNEM = Dataset.UNRATE; Y = [CPID,UNEM(2:end)];
Create a VAR model:
Spec = vgxset('n',2,'nAR',4,'Constant',true) Spec = Model: 2-D VAR(4) with Additive Constant n: 2 nAR: 4 nMA: 0 nX: 0 a: [] AR: {} Q: []
Fit the model to the data using vgxvarx:
[EstSpec,EstStdErrors,LLF,W] = vgxvarx(Spec,Y); vgxdisp(EstSpec) Model : 2-D VAR(4) with Additive Constant Conditional mean is AR-stable and is MA-invertible a Constant: 0.00176552 0.313448 AR(1) Autoregression Matrix: 0.319541 -0.00296663 -4.21339 1.34922 AR(2) Autoregression Matrix: 0.190051 0.00109078 6.35142 -0.272057 AR(3) Autoregression Matrix: 0.359724 0.0027494 1.64566 -0.230127 AR(4) Autoregression Matrix: -0.0398043 -0.000907289 8.82158 0.078716 Q Innovations Covariance: 3.51871e-005 -0.000181614 -0.000181614 0.116329
Example: Fitting a VARMA Model. This example uses artificial data to generate a time series, then shows two ways of fitting a VARMA model to the series.
Specify the model:
AR1 = [.3 -.1 .05;.1 .2 .1;-.1 .2 .4]; AR2 = [.1 .05 .001;.001 .1 .01;-.01 -.01 .2]; Q = [.2 .05 .02;.05 .3 .1;.02 .1 .25]; MA1 = [.5 .2 .1;.1 .6 .2;0 .1 .4]; MA2 = [.2 .1 .1; .05 .1 .05;.02 .04 .2]; Spec = vgxset('AR',{AR1,AR2},'Q',Q,'MA',{MA1,MA2}) Spec = Model: 3-D VARMA(2,2) with No Additive Constant n: 3 nAR: 2 nMA: 2 nX: 0 AR: {2x1 cell} stable autoregressive process MA: {2x1 cell} invertible moving average process Q: [3x3] covariance matrix
Generate a time series using vgxsim:
YF = [100 50 20;110 52 22;119 54 23]; % starting values Y = vgxsim(Spec,190,[],YF);
Fit the data to a VAR model by calling vgxvarx with the 'IgnoreMA' option:
estSpec = vgxvarx(Spec,Y(3:end,:),[],Y(1:2,:),'IgnoreMA','yes');
Compare the estimated model with the original:
vgxdisp(Spec,estSpec) Model : 3-D VARMA(2,2) with No Additive Constant Conditional mean is AR-stable and is MA-invertible Model : 3-D VAR(2) with No Additive Constant Conditional mean is AR-stable and is MA-invertible Parameter Model 1 Model 2 -------------- -------------- -------------- AR(1)(1,1) 0.3 0.689078 (1,2) -0.1 0.106741 (1,3) 0.05 0.185514 (2,1) 0.1 0.055764 (2,2) 0.2 0.737816 (2,3) 0.1 0.379725 (3,1) -0.1 -0.134891 (3,2) 0.2 0.3787 (3,3) 0.4 0.776079 AR(2)(1,1) 0.1 -0.128783 (1,2) 0.05 -0.0815194 (1,3) 0.001 -0.0662361 (2,1) 0.001 0.0746446 (2,2) 0.1 -0.294495 (2,3) 0.01 -0.151349 (3,1) -0.01 0.0475872 (3,2) -0.01 -0.179413 (3,3) 0.2 -0.0415697 MA(1)(1,1) 0.5 (1,2) 0.2 (1,3) 0.1 (2,1) 0.1 (2,2) 0.6 (2,3) 0.2 (3,1) 0 (3,2) 0.1 (3,3) 0.4 MA(2)(1,1) 0.2 (1,2) 0.1 (1,3) 0.1 (2,1) 0.05 (2,2) 0.1 (2,3) 0.05 (3,1) 0.02 (3,2) 0.04 (3,3) 0.2 Q(1,1) 0.2 0.199153 Q(2,1) 0.05 0.0471438 Q(2,2) 0.3 0.328947 Q(3,1) 0.02 0.0141446 Q(3,2) 0.1 0.107757 Q(3,3) 0.25 0.243927
The estimated Q matrix is close to the original Q matrix. However, the estimated AR terms are not close to the original AR terms. Specifically, nearly all the AR(2) coefficients are the opposite sign, and most AR(1) coefficients are off by about a factor of 2.
Alternatively, before fitting the model, convert it to a pure AR model:
Specify the model as above.
Generate a time series using vgxsim as above.
Convert the model to a pure AR model:
specAR = vgxar(Spec);
Fit the converted model to the data:
estSpecAR = vgxvarx(specAR,Y(3:end,:),[],Y(1:2,:));
Compare the fitted model to the original model:
vgxdisp(specAR,estSpecAR) Model : 3-D VAR(2) with No Additive Constant Conditional mean is AR-stable and is MA-invertible Model : 3-D VAR(2) with No Additive Constant Conditional mean is AR-stable and is MA-invertible Parameter Model 1 Model 2 -------------- -------------- -------------- AR(1)(1,1) 0.8 0.689078 (1,2) 0.1 0.106741 (1,3) 0.15 0.185514 (2,1) 0.2 0.055764 (2,2) 0.8 0.737816 (2,3) 0.3 0.379725 (3,1) -0.1 -0.134891 (3,2) 0.3 0.3787 (3,3) 0.8 0.776079 AR(2)(1,1) -0.13 -0.128783 (1,2) -0.09 -0.0815194 (1,3) -0.114 -0.0662361 (2,1) -0.129 0.0746446 (2,2) -0.35 -0.294495 (2,3) -0.295 -0.151349 (3,1) 0.03 0.0475872 (3,2) -0.17 -0.179413 (3,3) 0.05 -0.0415697 Q(1,1) 0.2 0.199153 Q(2,1) 0.05 0.0471438 Q(2,2) 0.3 0.328947 Q(3,1) 0.02 0.0141446 Q(3,2) 0.1 0.107757 Q(3,3) 0.25 0.243927
The model coefficients between the pure AR models are closer than between the original VARMA model and the fitted AR model. Most model coefficients are within 20% or the original. Notice, too, that estSpec and estSpecAR are identical. This is because both are AR(2) models fitted to the same data series.
How vgxvarx Works. vgxvarx finds maximum likelihood estimators of AR and Q matrices and the a and b vectors if present. For VAR models, vgxvarx uses a direct solution algorithm that requires no iterations. For VARX models, the algorithm iterates. The iterations usually converge quickly, unless two or more exogenous data streams are proportional to each other. In that case, there is no unique maximum likelihood estimator, and the iterations might not converge. You can set the maximum number of iterations with the MaxIter parameter, which has a default value of 1000.
vgxvarx calculates the loglikelihood of the data, giving it as an output of the fitted model. Use this output in testing the quality of the model. For example, see Determining an Appropriate Number of Lags and Examining the Stability of a Fitted Model.
When you enter the name of a fitted model at the command line, you obtain a summary. This summary contains a report on the stability of the VAR part of the model, and the invertibility of the VMA part. You can also find whether a model is stable or invertible by entering:
[isStable,isInvertible] = vgxqual(Spec)
This gives a Boolean value of 1 for isStable if the model is stable, and a Boolean value of 1 for isInvertible if the model is invertible. This stability or invertibility does not take into account exogenous terms, which can disrupt the stability of a model.
Stable, invertible models give reliable results, while unstable or noninvertible ones might not.
Stability and invertibility are equivalent to all eigenvalues of the associated lag operators having modulus less than 1. In fact vgxqual evaluates these quantities by calculating eigenvalues. For more information, see the vgxqual function reference page or Hamilton [48]
When you have models with parameters (known or estimated), you can examine the predictions of the models. For information on specifying models, see Model Specification Structures. For information on calibrating models, see VAR Model Estimation.
The main methods of forecasting are:
Generating forecasts with error bounds using vgxpred
Generating simulations with vgxsim
Generating sample paths with vgxproc
These functions base their forecasts on a model specification and initial data. The functions differ in their innovations processes:
vgxpred assumes zero innovations. Therefore, vgxpred yields a deterministic forecast.
vgxsim assumes the innovations are jointly normal with covariance matrix Q. vgxsim yields pseudorandom (Monte Carlo) sample paths.
vgxproc uses a separate input for the innovations process. vgxproc yields a sample path that is deterministically based on the innovations process.
vgxpred is faster and takes less memory than generating many sample paths using vgxsim. However, vgxpred is not as flexible as vgxsim. For example, suppose you transform some time series before making a model, and want to undo the transformation when examining forecasts. The error bounds given by transforms of vgxpred error bounds are not valid bounds. In contrast, the error bounds given by the statistics of transformed simulations are valid.
Forecasting with vgxpred. vgxpred enables you to generate forecasts with error estimates. View the results with vgxplot. vgxpred requires:
A model (impSpec in the following example)
The number of periods for the forecast (FT in the following example)
vgxpred optionally takes:
An exogenous data series
A presample time series (Y(end-3:end,:) in the following example)
Extra paths
For details, see the vgxpred reference page.
This example uses two time series from Data_USEconModel, the logarithm of real GDP, and the real 3-month T-bill rate, both differenced to be approximately stationary. It uses a VAR(4) model, and has vgxvarx estimate the parameters.
Load the data and apply transformations:
load Data_USEconmodel DEF = log(Dataset.CPIAUCSL); GDP = log(Dataset.GDP); rGDP = diff(GDP - DEF); % Real GDP is GDP - deflation TB3 = 0.01*Dataset.TB3MS; dDEF = 4*diff(DEF); % Scaling rTB3 = TB3(2:end) - dDEF; % Real interest is deflated Y = [rGDP,rTB3];
Fit a VAR(4) model specification:
Spec = vgxset('n',2,'nAR',4,'Constant',true); impSpec = vgxvarx(Spec,Y(5:end,:),[],Y(1:4,:)); impSpec = vgxset(impSpec,'Series',... {'Transformed real GDP','Transformed real 3-mo T-bill rate'});
Predict the evolution of the time series:
FDates = datenum({'30-Jun-2009'; '30-Sep-2009'; '31-Dec-2009'; '31-Mar-2010'; '30-Jun-2010'; '30-Sep-2010'; '31-Dec-2010'; '31-Mar-2011'; '30-Jun-2011'; '30-Sep-2011'; '31-Dec-2011'; '31-Mar-2012'; '30-Jun-2012'; '30-Sep-2012'; '31-Dec-2012'; '31-Mar-2013'; '30-Jun-2013'; '30-Sep-2013'; '31-Dec-2013'; '31-Mar-2014'; '30-Jun-2014' }); FT = numel(FDates); [Forecast,ForecastCov] = vgxpred(impSpec,FT,[],... Y(end-3:end,:));
Plot the forecast:
Monte Carlo Forecasting with vgxsim. vgxsim enables you to generate simulations of time series based on your model. If you have a trustworthy model structure, you can use these simulations as sample forecasts.
vgxsim requires:
A model (impSpec in the following example)
The number of periods for the forecast (FT in the following example)
vgxsim optionally takes:
An exogenous data series
A presample time series (Y(end-3:end,:) in the following example)
Presample innovations
The number of realizations to simulate (2000 in the following example)
For details, see the vgxsim reference page.
This example uses the data and model from Forecasting with vgxpred.
Load the data, transform it, specify a model, and calibrate the model as in Forecasting with vgxpred.
Simulate the model for 10 steps, replicated 2000 times:
Ysim = vgxsim(impSpec,FT,[],Y(end-3:end,:),[],2000);
Calculate the mean and standard deviation of the simulated series:
Ymean = mean(Ysim,3); % Calculate means Ystd = std(Ysim,0,3); % Calculate std deviations
Plot the means ±1 standard deviation for the simulated series:
subplot(2,1,1) plot(dates(end-10:end),Y(end-10:end,1),'k') hold('on') plot([dates(end);FDates],[Y(end,1);Ymean(:,1)],'r') plot([dates(end);FDates],[Y(end,1);Ymean(:,1)]+[0;Ystd(:,1)],'b') plot([dates(end);FDates],[Y(end,1);Ymean(:,1)]-[0;Ystd(:,1)],'b') datetick('x') title('Transformed real GDP') subplot(2,1,2) plot(dates(end-10:end),Y(end-10:end,2),'k') hold('on') axis([dates(end-10),FDates(end),-.1,.1]); plot([dates(end);FDates],[Y(end,2);Ymean(:,2)],'r') plot([dates(end);FDates],[Y(end,2);Ymean(:,2)]+[0;Ystd(:,2)],'b') plot([dates(end);FDates],[Y(end,2);Ymean(:,2)]-[0;Ystd(:,2)],'b') datetick('x') title('Transformed real 3-mo T-bill rate')
This figure differs from that in Forecasting with vgxpred in that the error bounds are zero for the period before simulation. The reason for the difference is that vgxsim takes the period before simulation as data, and has variability only in its predictions (simulations). vgxpred considers the period before prediction as arising from a random process, with normal innovations and covariance matrix Q. The pre-prediction standard deviation intervals are derived from Q. Otherwise, the two plots are strikingly similar. This validates that vgxpred gives accurate forecasts, since the forecasts closely match the results of a simulation with 2,000 replications.
How vgxpred and vgxsim Work. vgxpred generates two quantities:
A deterministic forecast time series based on 0 innovations
Time series of forecast covariances based on the Q matrix
The simulations for models with VMA terms uses presample innovation terms. Presample innovation terms are values of εt for times before the forecast period that affect the MA terms. For definitions of the terms MA, Q, and εt, see Types of VAR Models. If you do not provide all requisite presample innovation terms, vgxpred assumes the value 0 for missing terms.
vgxsim generates random time series based on the model using normal random innovations distributed with Q covariances. The simulations of models with MA terms uses presample innovation terms. If you do not provide all requisite presample innovation terms, vgxsim assumes the value 0 for missing terms.
If you scaled any time series before fitting a model, you can unscale the resulting time series to understand its predictions more easily.
If you scaled a series with log, transform predictions of the corresponding model with exp.
If you scaled a series with diff(log), transform predictions of the corresponding model with cumsum(exp). cumsum is the inverse of diff; it calculates cumulative sums. As in integration, you must choose an appropriate additive constant for the cumulative sum. For example, take the log of the final entry in the corresponding data series, and use it as the first term in the series before applying cumsum.
You can examine the effect of impulse responses to models with the vgxproc function. An impulse response is the deterministic response of a time series model to an innovations process that has the value of one standard deviation in one component at the initial time, and zeros in all other components and times. vgxproc simulates the evolution of a time series model from a given innovations process. Therefore, vgxproc is appropriate for examining impulse responses.
The only difficulty in using vgxproc is determining exactly what is "the value of one standard deviation in one component at the initial time." This value can mean different things depending on your model.
For a structural model, B0 is usually a known diagonal matrix, and Q is an identity matrix. In this case, the impulse response to component i is the square root of B(i,i).
For a nonstructural model, there are several choices. The simplest choice, though not necessarily the most accurate, is to take component i as the square root of Q(i,i). Other possibilities include taking the Cholesky decomposition of Q, or diagonalizing Q and taking the square root of the diagonal matrix.
Example: Calculating Impulse Responses with vgxproc. This example uses the data and model from Forecasting with vgxpred. It shows how to calculate and plot the impulse response of an interest rate shock on the real GDP.
Load the data, transform it, specify a model, and calibrate the model as in Forecasting with vgxpred.
Generate the innovations processes both with and without an impulse (shock):
W0 = zeros(FT, 2); % Innovations without a shock W1 = W0; W1(1,2) = sqrt(impSpec.Q(2,2)); % Innovations with a shock
Generate the processes with and without the shock:
Yimpulse = vgxproc(impSpec,W1,[],Y); % Process with shock Ynoimp = vgxproc(impSpec,W0,[],Y); % Process with no shock
Undo the scaling for the GDP processes:
Yimp1 = exp(cumsum(Yimpulse(:,1))); % Undo scaling Ynoimp1 = exp(cumsum(Ynoimp(:,1)));
Compute and plot the relative difference between the calculated GDPs:
RelDiff = (Yimp1 - Ynoimp1) ./ Yimp1; plot(FDates,100*RelDiff);dateaxis('x',12) title(... 'Impact of Interest Rate Shock on Real Gross Domestic Product') ylabel('% Change')
The graph shows that an increased interest rate causes a dip in the real GDP for a short time. Afterwards the real GDP begins to climb again, reaching its former level in about 1 year.
This section contains an example of the workflow described in Building VAR Models. The example uses three time series: GDP, M1 money supply, and the 3-month T-bill rate. The example shows:
Loading and transforming the data for stationarity
Partitioning the transformed data into presample, estimation, and forecast intervals to support a backtesting experiment
Making several models
Fitting the models to the data
Deciding which of the models is best
Making forecasts based on the best model
Loading Data. The file Data_USEconModel ships with Econometrics Toolbox software. The file contains time series from the St. Louis Federal Reserve Economics Database. This example uses three of the time series: GDP, M1 money supply (M1SL), and 3-month T-bill rate (TB3MS). Load the data into a time series matrix Y as follows:
load Data_USEconModel GDP = Dataset.GDP; M1 = Dataset.M1SL; TB3 = Dataset.TB3MS; Y = [GDP,M1,TB3];
Transforming Data for Stationarity. Plot the data to look for trends:
subplot(3,1,1) plot(dates,Y(:,1),'r'); title('GDP') datetick('x'), grid('on') hold('on') subplot(3,1,2); plot(dates,Y(:,2),'b'); title('M1') datetick('x'), grid('on') subplot(3,1,3); plot(dates, Y(:,3), 'k') title('3-mo T-bill') datetick('x'), grid('on') hold('off')
Not surprisingly, both the GDP and M1 data appear to grow, while the T-bill returns show no long-term growth. To counter the trends in GDP and M1, take a difference of the logarithms of the data. Taking a difference shortens the time series, as described in Transforming Data for Stationarity. Therefore, truncate the T-bill series and date series X so that the Y data matrix has the same number of rows for each column:
Y = [diff(log(Y(:,1:2))), Y(2:end,3)]; % Transformed data X = dates(2:end); subplot(3,1,1) plot(X,Y(:,1),'r'); title('GDP') datetick('x'),grid('on') hold('on') subplot(3,1,2); plot(X,Y(:,2),'b'); title('M1') datetick('x'),grid('on') subplot(3,1,3); plot(X, Y(:,3),'k'), title('3-mo T-bill') datetick('x'),grid('on') hold('off')
You see that the scale of the first two columns is about 100 times smaller than the third. Multiply the first two columns by 100 so that the time series are all roughly on the same scale. This scaling makes it easy to plot all the series on the same plot. More importantly, this type of scaling makes optimizations more numerically stable (for example, maximizing loglikelihoods).
Y(:,1:2) = 100*Y(:,1:2); figure plot(X,Y(:,1),'r');hold('on') plot(X,Y(:,2),'b'); datetick('x'), grid('on') plot(X,Y(:,3),'k'); legend('GDP','M1','3-mo T-bill'); hold('off')
Selecting Models. You can choose many different models for the data. This example rather arbitrarily chooses four models:
VAR(2) with diagonal autoregressive and covariance matrices
VAR(2) with full autoregressive and covariance matrices
VAR(4) with diagonal autoregressive and covariance matrices
VAR(4) with full autoregressive and covariance matrices
Load the data:
load Data_USEconModel GDP = Dataset.GDP; M1 = Dataset.M1SL; TB3 = Dataset.TB3MS;
Make the series the same length, and transform them to be stationary and on a similar scale.
dGDP = 100*diff(log(GDP(49:end))); dM1 = 100*diff(log(M1(49:end))); dT3 = diff(TB3(49:end)); Y = [dGDP dM1 dT3];
Create the four models as follows:
dt = logical(eye(3)); VAR2diag = vgxset('ARsolve',repmat({dt},2,1),... 'asolve',true(3,1),'Series',{'GDP','M1','3-mo T-bill'}); VAR2full = vgxset(VAR2diag,'ARsolve',[]); VAR4diag = vgxset(VAR2diag,'nAR',4,'ARsolve',repmat({dt},4,1)); VAR4full = vgxset(VAR2full,'nAR',4);
The matrix dt is a diagonal logical matrix. dt specifies that both the autoregressive matrices for VAR2diag and VAR4diag are diagonal. In contrast, the specifications for VAR2full and VAR4full have empty matrices instead of dt. Therefore, vgxvarx fits the defaults, which are full matrices for autoregressive and correlation matrices.
Choosing Presample, Estimation, and Forecast Periods. To assess the quality of the models, divide the response data Y into three periods: presample, estimation, and forecast. Fit the models to the estimation data, using the presample period to provide lagged data. Compare the predictions of the fitted models to the forecast data. The estimation period is in-sample, and the forecast period is out-of-sample (also known as backtesting).
For the two VAR(4) models, the presample period is the first four rows of Y. Use the same presample period for the VAR(2) models so that all the models are fit to the same data. This is necessary for model fit comparisons. For both models, the forecast period is the final 10% of the rows of Y. The estimation period for the models goes from row 5 to the 90% row. The code for defining these data periods follows:
Ypre = Y(1:4,:); T = ceil(.9*size(Y,1)); Yest = Y(5:T,:); YF = Y((T+1):end,:); TF = size(YF,1);
Fitting with vgxvarx. Now that the models and time series exist, you can easily fit the models to the data:
[EstSpec1,EstStdErrors1,LLF1,W1] = ... vgxvarx(VAR2diag,Yest,[],Ypre,'CovarType','Diagonal'); [EstSpec2,EstStdErrors2,LLF2,W2] = ... vgxvarx(VAR2full,Yest,[],Ypre); [EstSpec3,EstStdErrors3,LLF3,W3] = ... vgxvarx(VAR4diag,Yest,[],Ypre,'CovarType','Diagonal'); [EstSpec4,EstStdErrors4,LLF4,W4] = ... vgxvarx(VAR4full,Yest,[],Ypre);
The EstSpec structures are the fitted models.
The EstStdErrors structures contain the standard errors of the fitted models.
The LLF are the loglikelihoods of the fitted models. Use these to help select the best model, as described in Checking Model Adequacy.
The W are the estimated innovations (residuals) processes, the same size as Yest.
The specification for EstSpec1 and EstSpec3 includes diagonal covariance matrices.
Checking Stability. You can check whether the estimated models are stable and invertible with the vgxqual function. (There are no MA terms in these models, so the models are necessarily invertible.) The test shows that all the estimated models are stable:
[isStable1,isInvertible1] = vgxqual(EstSpec1); [isStable2,isInvertible2] = vgxqual(EstSpec2); [isStable3,isInvertible3] = vgxqual(EstSpec3); [isStable4,isInvertible4] = vgxqual(EstSpec4); [isStable1,isStable2,isStable3,isStable4] ans = 1 1 1 1
You can also look at the estimated specification structures. Each contains a line stating whether the model is stable:
EstSpec4 = ... AR: {4x1 cell} stable autoregressive process ...
Likelihood Ratio Tests. You can compare the restricted (diagonal) AR models to their unrestricted (full) counterparts using the lratiotest function. The test rejects or fails to reject the hypothesis that the restricted models are adequate, with a default 5% tolerance. This is an in-sample test. To perform the test:
Count the parameters in the models using the vgxcount function:
[n1,n1p] = vgxcount(EstSpec1); [n2,n2p] = vgxcount(EstSpec2); [n3,n3p] = vgxcount(EstSpec3); [n4,n4p] = vgxcount(EstSpec4);
Compute the likelihood ratio tests:
reject1 = lratiotest(LLF2,LLF1,n2p - n1p) reject1 = 1 reject3 = lratiotest(LLF4,LLF3,n4p - n3p) reject3 = 1 reject4 = lratiotest(LLF4,LLF2,n4p - n2p) reject4 = 0
The 1 results indicate that the likelihood ratio test rejected both the restricted models, AR(1) and AR(3), in favor of the corresponding unrestricted models. Therefore, based on this test, the unrestricted AR(2) and AR(4) models are preferable. However, the test does not reject the unrestricted AR(2) model in favor of the unrestricted AR(4) model. (This test regards the AR(2) model as an AR(4) model with restrictions that the autoregression matrices AR(3) and AR(4) are 0.) Therefore, it seems that the unrestricted AR(2) model is best among the models fit.
Akaike Information Criterion. To find the best model among a set, minimize the Akaike information criterion. This is an in-sample calculation. Here is how to calculate the criterion for the four models:
AIC = aicbic([LLF1 LLF2 LLF3 LLF4],[n1p n2p n3p n4p]) AIC = 1.0e+03 * 1.5129 1.4462 1.5122 1.4628
The best model according to this criterion is the unrestricted AR(2) model. Notice, too, that the unrestricted AR(4) model has lower Akaike information than either of the restricted models. Based on this criterion, the unrestricted AR(2) model is best, with the unrestricted AR(4) model coming next in preference.
Comparing Forecasts with Forecast Period Data. To compare the predictions of the four models against the forecast data YF, use the vgxpred function. This function returns both a prediction of the mean time series, and an error covariance matrix that gives confidence intervals about the means. This is an out-of-sample calculation.
[FY1,FYCov1] = vgxpred(EstSpec1,TF,[],Yest); [FY2,FYCov2] = vgxpred(EstSpec2,TF,[],Yest); [FY3,FYCov3] = vgxpred(EstSpec3,TF,[],Yest); [FY4,FYCov4] = vgxpred(EstSpec4,TF,[],Yest);
A plot shows the predictions in the shaded region to the right:
It is now straightforward to calculate the sum of squares error between the predictions and the data, YF:
error1 = YF - FY1; error2 = YF - FY2; error3 = YF - FY3; error4 = YF - FY4; SSerror1 = error1(:)' * error1(:); SSerror2 = error2(:)' * error2(:); SSerror3 = error3(:)' * error3(:); SSerror4 = error4(:)' * error4(:); figure bar([SSerror1 SSerror2 SSerror3 SSerror4],.5) ylabel('Sum of squared errors') set(gca,'XTickLabel',... {'AR2 diag' 'AR2 full' 'AR4 diag' 'AR4 full'}) title('Sum of Squared Forecast Errors')
The predictive performance of the four models is similar.
The full AR(2) model seems to be the best and most parsimonious fit. Its model parameters are:
vgxdisp(EstSpec2) Model : 3-D VAR(2) with Additive Constant Conditional mean is AR-stable and is MA-invertible Series : GDP Series : M1 Series : 3-mo T-bill a Constant: 0.687401 0.3856 -0.915879 AR(1) Autoregression Matrix: 0.272374 -0.0162214 0.0928186 0.0563884 0.240527 -0.389905 0.280759 -0.0712716 -0.32747 AR(2) Autoregression Matrix: 0.242554 0.140464 -0.177957 0.00130726 0.380042 -0.0484981 0.260414 0.024308 -0.43541 Q Innovations Covariance: 0.632182 0.105925 0.216806 0.105925 0.991607 -0.155881 0.216806 -0.155881 1.00082
This example shows two ways to make predictions or forecasts based on the EstSpec4 fitted model:
Running vgxpred based on the last few rows of YF.
Simulating several time series with the vgxsim function.
In both cases, transform the forecasts so they are directly comparable to the original time series.
Forecasting with vgxpred. This example shows predictions 10 steps into the future.
Generate the prediction time series from the fitted model beginning at the latest times:
[ypred,ycov] = vgxpred(EstSpec2,10,[],YF);
Transform the predictions by undoing the scaling and differencing applied to the original data. Make sure to insert the last observation at the beginning of the time series before using cumsum to undo the differencing. And, since differencing occurred after taking logarithms, insert the logarithm before using cumsum:
yfirst = [GDP,M1,TB3]; % the original data yfirst = yfirst(49:end,:); % NaN values removed dates = dates(49:end); endpt = yfirst(end,:); % the last time in the series endpt(1:2) = log(endpt(1:2)); % convert to log for cumsum ypred(:,1:2) = ypred(:,1:2)/100; % was multiplied by 100 ypred = [endpt; ypred]; % add final data point to beginning ypred(:,1:3) = cumsum(ypred(:,1:3)); % reason for adding endpt ypred(:,1:2) = exp(ypred(:,1:2)); % undoing the logarithm lastime = dates(end); timess = lastime:91:lastime+910; % extending the times subplot(3,1,1) plot(timess,ypred(:,1),':r') hold on plot(dates,yfirst(:,1),'k'),datetick('x'), grid('on') title('GDP') subplot(3,1,2); plot(timess,ypred(:,2),':r') hold on plot(dates,yfirst(:,2),'k'),datetick('x'), grid('on') title('M1') subplot(3,1,3); plot(timess,ypred(:,3),':r') hold on plot(dates,yfirst(:,3),'k'),datetick('x'), grid('on') title('3-mo T-bill')
The following plot shows the extrapolations in dotted red, and the original data series in solid black.
Look at the last few years in this plot to get a sense of how the predictions relate to the latest data points:
ylast = yfirst(170:end,:); timeslast = dates(170:end); subplot(3,1,1) plot(timess,ypred(:,1),'--r') hold on plot(timeslast,ylast(:,1),'k'),datetick('x'), grid('on') title('GDP') subplot(3,1,2); plot(timess,ypred(:,2),'--r') hold on plot(timeslast,ylast(:,2),'k'),datetick('x'), grid('on') title('M1') subplot(3,1,3); plot(timess,ypred(:,3),'--r') hold on plot(timeslast,ylast(:,3),'k'),datetick('x'), grid('on') title('3-mo T-bill')
The forecast shows increasing GDP, little growth in M1, and a slight decline in the interest rate. However, the forecast has no error bars. For a forecast with errors, see Forecasting with vgxsim.
Forecasting with vgxsim. This example shows forecasting 10 steps into the future, with a simulation replicated 2000 times, and generates the means and standard deviations.
Simulate a time series from the fitted model beginning at the latest times:
ysim = vgxsim(EstSpec2,10,[],YF,[],2000);
Transform the predictions by undoing the scaling and differencing applied to the original data. Make sure to insert the last observation at the beginning of the time series before using cumsum to undo the differencing. And, since differencing occurred after taking logarithms, insert the logarithm before using cumsum:
yfirst = [GDP,M1,TB3]; % the original data endpt = yfirst(end,:); % the last time in the series endpt(1:2) = log(endpt(1:2)); % convert to log for cumsum ysim(:,1:2,:) = ysim(:,1:2,:)/100; % undo the *100 ysim = [repmat(endpt,[1,1,2000]);ysim]; % insert endpt first ysim(:,1:3,:) = cumsum(ysim(:,1:3,:)); % undo diff ysim(:,1:2,:) = exp(ysim(:,1:2,:)); % undo log
Compute the mean and standard deviation of each series, and plot the results. The plot has the mean in black, with ±1 standard deviation in red.
ymean = mean(ysim,3); ystd = std(ysim,0,3); subplot(3,1,1) plot(timess,ymean(:,1),'k'),datetick('x'), grid('on') hold on plot(timess,ymean(:,1)+ystd(:,1),'--r') plot(timess,ymean(:,1)-ystd(:,1),'--r') title('GDP') subplot(3,1,2); plot(timess,ymean(:,2),'k'),datetick('x'), grid('on') hold on plot(timess,ymean(:,2)+ystd(:,2),'--r') plot(timess,ymean(:,2)-ystd(:,2),'--r') title('M1') subplot(3,1,3); plot(timess,ymean(:,3),'k'),datetick('x'), grid('on') hold on plot(timess,ymean(:,3)+ystd(:,3),'--r') plot(timess,ymean(:,3)-ystd(:,3),'--r') title('3-mo T-bill')
The series show increasing growth in GDP, moderate to little growth in M1, and uncertainty about the direction of T-bill rates.


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Bruck Lighting 100915MC/CM Matte Chrome / Blue Glass Smooth Blue Glass Ceiling Mount Light Fixture with Speckled Lighting Pattern from the Bobo Collection
Technical Specs
Specification Sheet | |
Collection | Bobo |
Depth | 4 |
Finish Application | Matte |
Height | 3.5 |
Installation Available | Yes |
Light Direction | Down Lighting |
Number of Bulbs | 1 |
Suggested Room Fit | Bedroom, Dining Room |
Theme | Modern, Retro |
Wattage | 50 |
Watts Per Bulb | 50 |
Width | 4.5 |
Configure your Smooth Blue Glass Ceiling Mount Light Fixture with Speckled Lighting Pattern from the Bobo Collection
Get It Installed more info ▼
When you order installation services from Installations Direct, a certified service provider in your community will contact you within 24 hours to make an appointment for your installation.
The service provider will install the new fixture and haul away the old fixture and any excess installation-related materials.
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Smooth Blue Glass Ceiling Mount Light Fixture with Speckled Lighting Pattern from the Bobo Collection
The glass of the Bobo CM is smooth on the outside but when illuminated will display speckled patterns. The Bobo CM includes a 4" ceiling mount canopy with 75 Watt transformer. Bobo with 2" ceiling mount canopy available.
- Includes Blue Glass Shade
About Bruck Lighting Systems:
Since its beginnings, Bruck Lighting Systems has operated under the principles of providing energy efficient lighting without compromising superior quality. With seven different track and cable systems, spot fixtures, display fixtures, decorative pendants, and sconces, low-voltage lighting has always been the focus.
As technology develops new energy efficient light sources, Bruck continues to design new and innovative products to utilize the latest lighting advancements. Two primary advancements that embody energy efficiency are Light Emitting Diodes (LEDs) and Compact Fluorescents. Bruck now offers an LED track system, multiple LED fixtures including some that are adapted for 12V track systems, LED pendants and sconces, and 18W compact fluorescent decorative pendants.
Additional Bruck Lighting Links
- Shop All Bruck Lighting Products
- Shop All Bruck Lighting Bobo Collection Products
Our SKU: 131915/CM
This product is listed under the following manufacturer number(s):
Bronze / Blue Glass - 131915CH/CM
Chrome / Blue Glass - 100915MC/CM
Matte Chrome / Blue Glass
*Denotes a finish or option that has been discontinued
Bruck Lighting
Bruck Lighting Bobo Collection
Other Products In The Bruck Lighting Bobo Collection
Bruck Lighting 320915
Line Voltage Pendant with 4" Mono-Point Canopy and Blue Glass Shade from the Bobo Collection
$234.00$206.25 -
Bruck Lighting 220915
Down Light Pendant with Smooth Blue Glass Shade from the Bobo Collection
$184.00$159.38 -
Bruck Lighting 320917
Line Voltage Pendant with 4" Mono-Point Canopy and Red Glass Shade from the Bobo Collection
$244.00$206.25 -
Bruck Lighting 103915
Up Light or Down Light Application 3 Watt LED Wall Sconce with Blue Glass Shade from the Bobo Collection
$294.00$271.88 -
Bruck Lighting K74001
Smooth Glass Shade with Speckle Pattern from the Bobo Collection


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News & Events
A three hour training session covering Foreclosure Settlement Conference practice will be given at 1 PM at the SCBA. The program is open to any practicing lawyer at no charge. Lawyers who volunteer for the Suffolk County Pro Bono Foreclosure Project and complete four appearance dates for the project will receive CLE credit for the program. This program is live only, it will NOT be webcast. Plese contact the SCBA, or click here to register on line. Registration is limited. Lunch will be served. The program is being underwitten by the Suffolk County Bar Association Pro Bono Foundation. The program is accredited by the Suffolk Academy of Law and is being presented jointly by the Suffolk County Bar Association and Nassau Suffolk Law Services. If you are currently a volunteer in the Foreclosure Settlement Project, completion of this program is required to remain listed as active on both the project and the LRIS Foreclosure Category.


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Vos économies : 34,52 $
Port et manutention : 6,99 $
(11 reviews)
Cet article est présentement en rupture de stock. Veuillez vérifier plus tard.
Pour obtenir de plus amples renseignements, veuillez composer le 1-888-2020-888.
Nous sommes désolés, le contenu de certaines pages n'est pas encore traduit en français. Il le sera bientôt.
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Great Collection of Fragrances!!!
Par Gayle the Shopper
de Winnipeg, Manitoba
I am in love with the Pheromone Musk fragrance, and decided to get the collection since it was on clearance. I have a few other favorites now, Sixth Sense,Pheromone Breeze,Fo Ti Tieng. I am still trying some others, very happy with this purchase :)
Love it
Par Aashi
de Edmonton Alberta
I am delighted with it! Such a wide variety of lovely scents at a such a great price. I love all of them the fragrance is super for all of them they in small bottle so even u can travel with them but just one thing they should come with a spray so its more esy to use but i Love itttttttttttt thanks for a great product
Par NANCY
de NANAIMO
I received this set today and am delighted with it! Such a wide variety of lovely scents at a such a great price. My favorite so far is the Sixth Sense, but I am looking forward to working my way thru them all. This would make a lovely gift for any woman, or a special treat to buy for yourself!
Love the collection
Par Stinkie
de Kitchener
Was very pleased with the collection. Great way to try all the scents and pick your favourites.
Fragrance longevity lacking
Par Ohkwari
de Surrey, BC
I was looking forward to receiving this collection based on the positive reviews here. I was however sadly disappointed as the scent does not stay on for more than a couple of hours. I was expecting higher quality in terms of staying power. I returned it.
( 1 of 1 clients ont trouvé ce commentaire utile)
Great collection!
Par Emmicris
de Toronto
Love the variety of subtle, attractive scents. Bonus: they don't cause an allergic reaction or give me a headache.
a delightful surprise!
Par Liz
de Brandon, Manitoba
This collection was a delightful surprise for me, did not expect such high quality. Each scent is unique in it's own way and it is such fun having such a variety of scents to choose from every day. Thank you to Marily Miglin for offering this wonderful set.
A Wonderful Treat!
Par Sunshine
de Calgary
I purchased these as sort of a Boxing Day gift to myself. The packaging is lovely, and I'm making my way through the scents slowly, savouring them like a box of chocolate! I love it! (My husband does too...) I don't have a favourite scent yet, but I like them all so far. Nice and light, not overpowering.
( 4 of 4 clients ont trouvé ce commentaire utile)
Totally in love with this kit
Par Samantha
de Markham, Ontario
I have to say that I am in full agreement with the reviewer who picked "Sixth Sense" as her favorite. It is truly a unique scent. Destiny Blue is another one that stands out. I have not tried them all but I must say that Marilyn's fragrances are all SO SPECIAL. So far I have not come across any that I find objectionable. All the pheromones fragrances are exquisite! Please keep them coming!
( 6 of 6 clients ont trouvé ce commentaire utile)
Hava Hava Elegant of Senses
Par judith
de Winnipeg Manitoba
for modern women with a tuch of old time so happy that i have bought it.one of best gift for every one any time of the year/.


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'The Sound of Music Live!'
For the first time in more than 50 years a broadcast network will air a full musical on live TV.
Carnegie Mellon University student Michael Campayno (A'14) and Tony Award-winning alumnus Christian Borle (A'95) are among the actors in "The Sound of Music Live!" which will be on NBC at 8 p.m. (EST), tonight.


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An attorney representing the estate of a two-month-old baby who died from a viral infection of the heart told the state Supreme Court on Wednesday that the trial court's jury instruction discussing the "error in judgment" rule reflected precarious legal terrain.
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Justices Mull 'Error in Judgment' Rule in Med Mal Cases
The Legal Intelligencer
November 29, 2012
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Fraser Coast Top Tourist Park Reviews
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The rating percentage is calculated from customers who booked within the last 12 months.
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Very friendly staff. Good size room - some rooms had a lot of wasted space though. Mirrors in rooms would be nice! Nice cap kitchen - always clean Spa and Jacuzzi-perfect! perfect location for urungan pier Great café attached with breakfast deals
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Culture and diversity revealed in archives
09 August
The Archive Awareness Campaign unveils its Culture and Diversity theme today, inviting the public to celebrate culture, diversity, the arts and creativity. The campaign is honouring local artists who have made an impact on the history of our communities, and also brings to light stories and artefacts from the places, people and cultures that make up the UK today.
Festival of Britain
Coinciding with the 60th anniversary of The Festival of Britain, Culture and Diversity: What's Your Story? has been inspired by the Festival's attention to the arts, design, architecture and culture of post-war Britain. The Festival was designed to be a celebration of all things British, both historic and contemporary. It represented the varied communities and social and economic classes involved in the celebrations.
Angela Owusu, of Archive Awareness Campaign, said: 'As part of this year's campaign, archives across the country are hosting special events to celebrate the fantastic wealth of their collections. Diverse activities are being held all over, from exhibitions, talks and competitions to an archive feast where the public will have the chance to taste a Regency-era seed cake, which once doubled as a cure for madness!'
Regional highlights include:- Norfolk Record Office will be holding a twilight event where visitors can participate in a live exhibition and view footage of the Festival of Britain celebrations. The Record Office will also explore the 1950s celebrations of the Queen's coronation
- the Victoria and Albert Museum is staging Inside Outing, which will involve an open day with a morning of speakers and an afternoon tour at the Southbank's Museum of 1951, plus an exhibition utilising its collection of Festival of Britain material
- a collection of previously forgotten recipes and foods from across the generations and cultures in Scotland will be tried and tested at the Scottish Council on Archives
- Cornwall Record Office will be launching a series of exhibitions looking at Cornish culture, its arts, festivals, and lifestyle
- the Irish Traditional Music Archive will be holding an exhibition honouring traditional Irish music on long play. The exhibition will allow people to handle LPs as well as listen to rare recordings
Commenting on the aim of the campaign, Geoff Pick of the Archives and Records Association (UK and Ireland) said: 'This campaign offers everyone the opportunity to explore the rich resources and discover the fascinating quality of the UK's cultural heritage. It is paying homage to the arts, contemporary culture and the hidden stories of cultural groups across the country.'
Culture and Diversity: What's Your Story? events will take place in communities across the country.


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Teen Ink
You have an interesting idea. An interesting flow. But its so jumpy and confusing, I don't know what's happening. I thought she was running, but then her friends weren't tied up, and were hitting their heads on windows? And somehow she knew what the killer was thinking, and that he was a quiet kind of person, and she had seen him before? If your bringing us into the middle of the action, we need a lot better character developement, and a lot better explained action. You can't just... (more »)
You mean like all those heated debates on my other reviews? Everyone else, even the people that I'm harsher on, haven't taken this so personally. I understand that writing is your passion, but you can't say it deosn't need improvement just because you love it the way it is. It needs improvement, I'm not debating about.
Can I point out that very few people on TeenInk actually review things? And currently, I'm the only one volunteering without requiring reciprocation? Maybe t... (more »)
Usually I don't really read this sort of thing, but today has been an exception.
This is AMAZING!!... I saw your thread about reading only two pages and that was my intention but I couldn't help but continue reading this wicked novel.
You so need to write part three & continue the story!...
I particularly like the way you use repetition of words, it familiarises the reader with the text and allows for them to be reminded and recall previous parts that they... (more »)


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which of the above songs made you become a fan, or fall in love with her music, or think "Wow, that girl is good!"?
for me it was Complicated.
if it was not a single that grabbed you but a regular song (like Things I'll Never Say, or Forgotten or something),tell which one it was. i made a "none of these" option for that reason


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Painted mural in Chemainus
Sightseeing (Picture BC photo)
For information on sightseeing tours in this location and other nearby locations, check out the map and product listings from business or service providers below.
Local Visitor Centres may also be able to offer helpful information about sightseeing tours available in this area.
4 Sightseeing Tours in Chemainus
Ageless Adventure tours, formally known as Mandate Tours, is an established BC-owned & operated tour company with over 24 yrs experience. We specialize in escorted tours throughout Canada & the US. In British Columbia, our tours explore various regions including Haida Gwaii, Thompson-Okanagan, Sunshine Coast, Cariboo, Northern BC & the Vancouver Island. Our clients can expect diverse sightseeing, comfortable accommodation, quality meals & tours that provide an historic perspective. Each tour is led by a professional & personable tour guide who will create a fun, casual atmosphere. Call us for our free brochure. We can customize group travel.
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Discover the story behind Vancouver Island's ancient rainforests of giant trees, alpine flower meadows & turquoise mountain lakes; ocean tide pools; marine & other wildlife; salmon & bird watching areas. Year-round personal nature tours with a biologist trained by Interpretation Canada. Get to the right place in less time so you can spend more time exploring, learning, exercising & experiencing. Find secret places & special moments off the beaten path & discover the hidden creatures others don't know about. Have an authentic vacation that tells the story behind the scenery through thought-provoking & interactive programs. Snowshoeing & hiking tours.
Tracks Outdoor Adventures INC is locally owned & operated in Nanaimo, Vancouver Island since 2002. Offering sustainable & local tours to all destinations on Vancouver Island including~ Local Wine/ Cider/ Micro Brew & distillery tours, nature & walking tours, sightseeing tours, heritage tours, culinary experiences touring groups from 4-500! We cater to the individual travellers, corporate & conference groups, cruise ship shore excursions. We also can customize any tour for you. Private group tours, shuttle services & Step on tour guide services are also available. Call to let us create a tour for you! Happy touring! Paula CEO Tracks INC.
A trip to British Columbia is not complete without a cruise on BC Ferries. Sail through spectacular scenery aboard our modern fleet of vessels with 25 routes throughout coastal British Columbia, including 3 connecting Vancouver to Vancouver Island. Circle Routes are a natural using BC Ferries with service to the Sunshine Coast & the Gulf Islands. Iconic cruises include the 15 hr Inside Passage between Port Hardy & Prince Rupert, as well as the Discovery Coast Passage between Port Hardy & Bella Coola & the mystical Haida Gwaii between Port Hardy & Skidegate. Visit bcferries.com for travel deals & itinerary suggestions.
Travel Deal


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Hyphenated - Americans | Rate Topic |
Author | Post |
You have chosen to ignore Bama46. click Here to view this post |
Posted: Wed Feb 6th, 2008 03:39 pm |
2nd Post |
susansweet Member
Okay Ed, I am a Welsh-Irish -Scots-Choctaw, Cherokee-English- Confederate-Yankee-Western American. So I am first and last an American. |
You have chosen to ignore Bama46. click Here to view this post |
Posted: Wed Feb 6th, 2008 07:49 pm |
4th Post |
Marie Member
I'm English-Irish-Shawnee-French-Swiss-German (Alsace-Lorraine-think Amish)- Ohioan by way of Canada, Pennsylvania & Virginia. First, last and always proud to be an American. Regards, Jana |
You have chosen to ignore Bama46. click Here to view this post |
Posted: Wed Feb 6th, 2008 11:20 pm |
6th Post |
Johan Steele Life NRA,SUVCW # 48,Legion 352
I'm just a Heinz 57 w/ a little of everything in me. I'm fluent in several languages: English, Bad English and Drunkanese. I know enough Turkish, Lakota, French and Spanish to get myself to the bathroom and a beer... we won't talk about my Russian. There is Swiss, German (Hessian), French, English, Native American (don't know what tribe), Czech, Tibetan and probably a mess more. Now my daughter has all of the above plus Canadian, French, Greek, Mohawk, African (we've no idea what)... so what does that make her? American cause if anyone calls her a mutt we'll have a long and probably uncomfortable conversation. Pure and simple my ancestroy doesn't mean spit; I'm an American. I'm not likely to go to, let's say, Switzerland and look up relatives; what would I say? My lines been over in the US for a century, wanna catch up on old times? Last edited on Wed Feb 6th, 2008 11:24 pm by Johan Steele |
Posted: Wed Feb 6th, 2008 11:53 pm |
7th Post |
JoanieReb Member
"But I speak two languages...English and Southern!" |
Posted: Thu Feb 7th, 2008 01:04 am |
8th Post |
CleburneFan Member
Here's a hyphenated name to consider. My elder son was born in Nairobi, Kenya. Hubby and I are American citizens. At eighteen, my son had to declare somewhere whether he was Kenyan citizen or United States citizen. He chose US citizen and registered for the US draft. Nothing can take away the fact that he was born in Kenya. My younger son was born in Portugal. My younger son has never, ever called himself a Portuguese-American nor has my elder son ever called himself African-American or Kenyan-American. One time I did make a blunder of mentioning that my elder son is an African-American, but in a special sense. The person to whom I was speaking took great umbrage with me and said my remark was in very poor taste. I asked why. She explained to me that an African-American is a person of African descent who is born in the United States and I was presumptuous, even racist to claim that because my son was born in Africa, but now lives in the US and is a US citizen that he is African-American. She was very angry with me and never spoke to me again. I was not claiming for my son the rights or privileges of African-Americans in any way such as using his status to try to get him admitted to a university or any such thing. But what happened that time was to show me how sensitive the issue of claiming a hypenated status is. It also shows how complicated the issue is. I find it too bad that my sons cannot make any claim whatsoever to having been born in another country. In fact, they do seldom tell anyone because it invariably involves long explanations. What this woman was trying to make clear to me is that my son does not claim the right to be African just because he was born there. She probably meant that those who claim a hypenated heritage are referring to the ethnic and even racial traits of those designations. Being born in a place isn't enough. Still, I wish we in the US wouldn't hypenate our nationality. I think it is divisive. Last edited on Thu Feb 7th, 2008 01:05 am by CleburneFan |
Posted: Thu Feb 7th, 2008 01:28 am |
9th Post |
JoanieReb Member
I know that we are supposed to be keeping it light, but I just wanted to say, Thank You, Cleburne Fan for this statement: "Still, I wish we in the US wouldn't hypenate our nationality. I think it is divisive." (I actually have ancestry for a serious brag, but let that go a long time ago, felt boxed in by it, never mention it anymore. Oops, guess I just did... but I ain't tellin' what. Anyway, they earned it, not me. I just got their genes). Now back to keeping it light! |
Posted: Thu Feb 7th, 2008 02:38 am |
10th Post |
CleburneFan Member
I'll make it heavy again, Joannie Reb. I have just recently found out that I have ancestors who were tried in the Salem Witch Trials. WOW! Is that a heavy trip or what? Maybe my hyphenated designation should be Accused Witch-American. |
Posted: Thu Feb 7th, 2008 02:41 am |
11th Post |
JoanieReb Member
Hee-hee - I LIKE it! "Accused Witch American", don't hear that everyday! |
Posted: Thu Feb 7th, 2008 02:45 am |
12th Post |
Dixie Girl Southern Belle
i am part Cherokee Southern and just plain American as far as i know whatever else i am im proud of it Last edited on Thu Aug 28th, 2008 06:40 pm by Dixie Girl ____________________ War Means Fighting And Fighting Means Killing - N. B. Forrest When war does come, my advice is to draw the sword and throw away the scabbard." Stonewall Jackson |
Posted: Thu Feb 7th, 2008 03:01 am |
13th Post |
CleburneFan Member
JoanieReb wrote: Hee-hee - I LIKE it! "Accused Witch American", don't hear that everyday! It surely does explain a lot! I really would have rather been a Cleburne-American or something equally cool. |
Posted: Thu Feb 7th, 2008 04:29 am |
14th Post |
JoanieReb Member
Cleburne Fan, Now that's a fun idea: Come up with our fantasy-hyphenated-selves! |
Posted: Thu Feb 7th, 2008 04:44 am |
15th Post |
Fuller E Pluribus Unum
Cleburne, maybe our ancestors were neighbors! My direct Fuller line left England for Salem and Rehoboth in 1638. No traces of the witch trials on their backs but lots of other history concerning them I know. As for me, I'm mostly English with some Welsh, German, Scottish, Swiss, Danish, Swedish and Norwegian. That translates to "whiter than Wonder bread" and I keep the spf 500 lotion handy at all times. I vote here and I pay taxes here. There has never been any hyphenating for me. |
Posted: Thu Feb 7th, 2008 04:59 am |
16th Post |
susansweet Member
Fuller ,one side of my family arrived in Roxbury in 1636 so maybe they were neighbors of yours at some time. Since all but the Indian ancestors were from the British Isles I also keep the SP 500 handy, I have brown hair but fair skin. Burn in two seconds outside. I forgot to add my great grandfather was born in Canada so add Canadian to the list . But then his parents were from New England area and oh yeah he was a Union Soldier in the Civil War. Bama I speak Californian and English. Cowabunga dudes and dudettes. Johan I have learned enough Danish in my life to order food , and tell my Danish kids to sit down, be quiet, eat your food, and ask the dog if it needs to go out side. Living in Southern Cal and after taking two years in High school and one year in college I can speak a little spanish , mainly order off the menu in any Mexican restaraunt. Susan |
Posted: Thu Feb 7th, 2008 01:16 pm |
17th Post |
Johan Steele Life NRA,SUVCW # 48,Legion 352
I think one thing is clear; all of the members of this board are History-American... or maybe Hysterical-American. |
Posted: Thu Feb 7th, 2008 03:02 pm |
18th Post |
Fuller E Pluribus Unum
Johan, crazy people have more fun dontcha think? Susan, I'm assuming you have been to Solvang right? I love that place. Aaaaaand, I just learned how to make Aebleskivers! Yum. |
Posted: Thu Feb 7th, 2008 03:37 pm |
19th Post |
CleburneFan Member
Johan Steele wrote: I think one thing is clear; all of the members of this board are History-American... or maybe Hysterical-American. Gotta love it--History-American and Hysterical-American. I am proudly and unabashedly both!!! |
Posted: Thu Feb 7th, 2008 04:04 pm |
20th Post |
susansweet Member
Fuller yes I have been to Solvang, but having been to Denmark I always found it too touristy . Usually too I was headed to Danish friends home so got better Danish food there than at Solvang. I love it History-American , that's me . We are all Book-Americans too or Reader-Americans. Susan |
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Civil War Interactive Discussion Board > The Lounge > Idle Chit-Chat > Hyphenated - Americans | Top |


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DNA testing finds evidence of widespread seafood fraud
Bait and switch: buyer beware?
Apr 23, 2012
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Bait and switch: buyer beware?
Apr 23, 2012
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It is something fishy going on at your local seafood market or restaurant customers are willing to pay premium prices for the best catch of the day.
But new DNA testing finds widespread.
Fraud it is bait and switch you might say and consumers are on the hook.
We -- us live in San Pedro California.
William what's gone.
Well John this is one -- one way to at least know what you're getting but it's no way to fight seafood fraud.
Of consumers Los Angeles about 40% in Boston Paper one fish -- got another for instance can you tell the difference between.
-- -- -- -- -- -- -- -- A soul and a sea bass or in this case a wild salmon in the steelhead Trout difference twenty dollars a pound probably not.
And many retailers are banking on.
Police found was quite alarming we found that more than half of the -- that we samples from restaurants sushi bars and grocery stores were mislabeled substituting one species of fish for another.
Experts recommend eating seafood twice a week but a new report says were giving scam I get dishonesty people are looking a problem here expensive.
An educator the end knowledgeable consumer report by advocacy group Oceana tested seafood around Los Angeles and found consumers buying one fish but getting another but as a -- drive like a Cadillac.
No it doesn't there are distinct differences the report found these fish most commonly mislabeled.
Two floppy or rock fish for snapper perch are Pollock for grouper make -- shark for swordfish.
Ask a liar which causes stomach problems instead of white tuna cat fish for Dover sole and flounder or sea bass for how -- It doesn't start with the Fisher usually it doesn't start with the -- those that are trying to make their living.
I'm the oceans but it's usually somewhere towards the end of that chain that is seeing an opportunity to take a product.
Similar to another product and repackage it.
Santa -- -- created a fish to -- tracking system.
Oceana recommends a similar federal Paper trail for all seafood sold in the US from trawler to tape.
We sort of have a patchwork.
Of enforcement and regulation and and none of it is really.
-- the level that we think is needed.
So in this study were 74 restaurants and -- in LA tested the worst for sushi restaurants 87% Broadway John.
Of the -- that was -- and was wrong.
Experts say number one go to reputable fish market number two of the price looks too good they're probably -- -- -- collage and S way and catching yourself.
Thanks William.
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Ororama is one of the original super store in the city. It has a supermarket, pharmacy and department store. Also has Bingo and Lotto outlets, mail & airline/ferry ticketing booth, as well as lots of cellphone and food tenants including Jollibee, McDonalds, Mang Inasal.
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TJ Cunningham
Sunday, September 1 12AM – Monday, September 30 12AM at Edgewater Gallery in Middlebury.
Contact Info:
- 802-458-0098
Seven Days Says:
"Adirondack Lake," paintings of the Vermont artist's recent hike through the Cascade Lake area in New York.
TJ Cunningham
Vermont painter TJ Cunningham recently strapped on a 60-pound backpack — filled with taboret and easel, among more conventional hiking gear — and hiked into the Adirondack Park. His goal? To sketch the landscape of the Cascade Lake region and to bring a bit of that wildness back to his studio. At 25, Cunningham seems to be preternaturally talented, producing rich and poetic oil paintings. The lush landscapes that resulted from his hike are on display in a show called "Adirondack Lake" at Edgewater Gallery in Middlebury through September 30. Pictured: "Birch."
- Reception: TJ Cunningham [Art]
Friday, September 13, 5–7PM at Edgewater Gallery in Middlebury.


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Cassell, Mark and Mullaly, Sarah. "When Smaller Governments Open the Window." State and Local Government Review. Accepted for publication.
Claassen, Ryan. 2011. "Political Awareness and Partisan Realignment: Are the Unaware Unevolved?" Political Research Quarterly 64: 818-830.
Claassen, Ryan. "Voter Confidence and the Election-Day Voting Experience," Political Behavior. Accepted for publication.
Coy, Patrick. "Nonpartisanship, Interventionism and Legality in Accompaniment," The International Journal of Human Rights. Accepted for publication.
Ensley, Michael. "Who Fits the Left-Right Divide? Partisan Polarization in the American Electorate," American Behavioral Scientist. Accepted for publication.
Ensley, Michael. "Incumbent Positioning, Ideological Heterogeneity, and Mobilization in the U.S. House," Public Choice. Accepted for publication.
Ensley, Michael. Robert J. Dole Archive Travel Grant ($700) from the University of Kansas.
Hawes, Daniel and Hill, Gregory C. 2011. "Managing Undocumented Students: Do Undocumented Students Hinder Student Performance?" State and Local Government Review 43: 183-195.
Hook, Steven W. ed., with Jones, Christopher. The Routledge Handbook of American Foreign Policy. New York: Taylor & Francis , 2012.
Handcock, Landon. "Transitional Justice and the Consultative Group," Ethnopolitics. Accepted for publication.
Manoharan, Aroon. "A Study of the Determinants of County E-Government in the United States," American Review of Public Administration. Accepted for publication.
Stacher, Joshua. Adaptable Autocrats: Regime Power in Egypt and Syria. Palo Alto: Stanford University Press, 2012. Release date April 5.