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# Volatility Modeling - Science topic

Explore the latest questions and answers in Volatility Modeling, and find Volatility Modeling experts.
Questions related to Volatility Modeling
Question
I want to investigate cross market volatility spillover among different indices. E-Views only allows investigate return and own spillover, not the cross-market spillover. So How can I perform the cross volatility spillover in R or any other software
Derek Cheng
Thank You it helped a lot
Question
I wish to transform the data from long memory stochastic to short memory volatility models to visualize the effects. Is there any transformation protocol to shift the data from long memory stochastic to short memory volatility models or either there is no mathematical relationship between these two models? Further, can we use MATLAB for such transformation?
Long memory volatility data are best modelled as FIGARCH models. There are perhaps no known ways of transformation from long memory to short memory data.
Question
In time series modeling and volatility estimation it is necessary, first remove autocorrelation of time series and after that estimate the volatility model (like GARCH).
the autocorrelation estimate by ACF test, but in some situations (like a low sample data or noise,...) maybe this procedure causes bad estimation of autocorrelation.
for example the true model is AR(3)-GARCH(1,1) but we used AR(1)-GARCH(1,1)
are the GARCH parameters biased in this situation?
The estimated model AR(1)-GARCH(1,1) is different from the true model AR(3)-GARCH(1,1). The estimates must therefore be biased.
Question
I'd really appreciate it if you could perhaps talk about how the topic relates to stock market efficiency and investment decisions.
Question
In my work, I use the multivariate GARH model (DCC-GARCH). I am testing the existence of autocorrelation in the variance model. Ljung-Box tests (Q) for standardized residuals and square standardized residuals give different results.
Should I choose the Ljung-Box or Ljung-Box square test?
N=1500
The Ljung-Box test is aimed at testing the independance of errors using residuals of an ARMA model estimated on the same data. But it makes use of autocorrelations so it is not powerful when the errors are uncorrelated but not independent. When applied to squared residuals, it can reveal ARCH and GARCH effects. Note that the errors of a ARCH-GARCH model are uncorrelated but not independent. Have a look at the excellent book by Francq and Zakoian entitled "GARCH Models: Structure, Statistical Inference and Financial Applications" published by Wiley in 2010.
Question
I am very keen to join the Post Doctoral Fellowship programme in economics. My area of specialisation is macroeconomics, food inflation, development and volatility modelling. I am looking for this position in Asian and Australian continent.
Have you looked at the Job Openings for Economists website? Many of the deadlines may have passed, but they may still accept for post-doc.
Question
Dear Madam, Would you like to help me to flag out the language or code for Stochastic volatility of GARCH model using high, low and close price of stock data.
Hi,
Faisal Nawaz Please refer to the article at:
Question
.
There is no single, universal, multi-faceted model built of many indicators selected from fundamental and technical analysis, including market, economic and financial analyzes, indicator analyzes, due diligence and others that could be used to analyze the situation, economic situation, volatility, etc. for every capital market, including the currency market, stock market, market of raw materials and other production factors, and for every situation, i.e. bear market, bull market, balance, strong changes in trends, different levels of investment risk, various phases in the economic cycle of the economy, and irrespective of the national specifics of the operation of financial markets. Specific analytical models should be built for a specific economic situation, for a specific market. Then it is possible to achieve a high level of decision-making accuracy on the basis of forecasts formulated from this type of multi-faceted, complex indicator models for analyzing the current and prospective situation on a specific stock exchange market or other specific capital market operating in a specific macroeconomic environment, in a specific financial system, in a specific financial system country.
Greetings.
Dariusz Prokopowicz
Question
It has been found in the existing volatility literatures that there is lacking in use of robust methodology while selecting time duration of empirical work. In most of the cases, either it has been decided based on the availability of the data or otherwise on an adhoc basis.
It is a broad issue and there is no a single answer. It depends on what is the aim of your research. One important parameter is what are the major developments and events during the time of research. Are there breakpoints? What is the frequency of data? There is always a minimum of observations. I think you should test using different duration and after comparing results, you will be able to choose the optimal solution. Experience in modeling is a major factor of success.
Question
Dr. Yaya, please, I wish to know the status of the work on "Volatility Modelling using Daily and Intraday High Frequency Datasets"
It is a general project area not a particular paper.
Question
I am taking multiple indices and I want to get cross countries analysis
This Matlab toolbox might help:
Question
is there any difference between exponential moving average(EMA) and exponentially weighted moving average(EWMA)?
The exponential smoothing method gives greater weight to the more new observations, while moving average give equal weight to all observations
Question
As I know, RiskMetrics uses lambda value of 0.94 to compute EWMA.
But, it is assigned arbitrarily.
Is there any method to estimate lambda value, instead of taking its value arbitrarily?
Definitely, lambda can be estimated within maximum-likelihood framework. Refer to the page 507 in Tsay (2010) for the log-likelihood structure.
Tsay (2010). Analysis of financial time series. John Wiley.
Question
Hi,
I have developed several volatility models for international arrivals from several countries using SARIMA-GARCH and SARIMA-GJR methods assuming a normal distribution in the estimation process. However, when I checked the normality of the residuals of each of the models, majority of the residuals are not normally distributed. So now I have a few questions.
1.Is normality of residuals a must in this context? If so how can I make it normally distributed?
2. Can I assume a student-t distribution or GED distribution in the estimation process allowing fore more flexibility in the model? or is there any other suitable method?
I used Eview 10 for estimations.
Much appreciated if anyone could advise in this regard.
Kind regards
Thushara
Yes, that is right. This what one does in practice. EViews output returns three info crit: AIC, SIC, and Hannan Quinn.
Question
As we know implied volatility is derived by interpolation of market price and the guess of the volatility by using the option pricing formula.
what are the real life applicaiton of implied volatiltiy
An option trader needs the implied volatility, at least of liquid options, not to price, the price is given by the market, but to hedge the options, by means of the underlying assets and a proxy of a bankaccount. I advise you, to pay a visit to option traders, in order to get a picture of what is going on in the trading pit!
Question
Iam trying to value electricity forward contract from the spot price model using Heston stochastic volatility model for short term contract like weekly. I also intend to price spark spread options out of this model. Incompleteness of markets and partial hedging problems are some of the challenges of this area of research that is why it has limited literature, anyone with links or suggested literature that answers some of the challenges and how to fully implement the closed form solution in Matlab or Mathematica
Volatility modelling may be by generalized autoregressive conditional heteroscedasticity (Garch) modelling in which case the conditional variance is modelled. The autocorrelation function of the squares of the data may be used as a diagnostic tool in this regard.
Question
This question come under regime switching volatility modeling of time series data. Using regime switching, by definition, invalidates any talk of stationarity in my opinion. What do financial economists think about this? Thank you.
I have to clarify myself a bit here. In any analysis of time series data, we always ensure the series is stationary before anything else. Now in regime-switching models, is that necessary? I hope that makes the question clearer.
Question
Seeking studies involving applications of the GARCH-MIDAS model.
The GARCH-MIDAS applications are gaining ground in the recent literature.
See, for example, among others:
Wei et al. (2018). Hot money and China’s stock market volatility: Further evidence using the GARCH–MIDAS model. Physica A: Statistical Mechanics and its Applications.
Pan et al. (2017). Oil price volatility and macroeconomic fundamentals: A regime switching GARCH-MIDAS model. Journal of Empirical Finance.
Question
I am currently carrying out an assignment in which we need to compare our GARCH( p, q) forecast against a Naive Benchmark model, such as a Random Walk model.
I have considered an IGARCH (1, 1) model to fall under this defintion, due to the coefficents of alpha and beta adding to 1.
Is this therefore a random walk with a drift model, as I am struggling with its interpretation?
I have attached a sample of the Eviews output for reference.
Hi: the random walk equation for sigma_t is
sigma_t = sigma_t-1 + epsilon_t.
so the coefficient of the previous sigma value would be +1.0 if RW for volatility is true.
the "naive model" versus igarch ) is probably that volatility follows a random walk. so, besides what the other people said about forecasting comparisons, another way is to check if volatility follows a random walk by estimating the model above and seeing if the coefficient on sigma_t-1 is +1.
actually, both methods should probably be done. but, the forecasting procedure should be done on out of sample observations that are not part of the respective estimation procedures. so, maybe split the data into two parts and estimate using half and then forecast using the other half. I hope this helps.
Question
Through the use of Eviews, I have carried out both a dynamic and static forecast for my GJR-GARCH (1, 1)- MA(1) model.
However, I am struggling to interpret these results and am unsure if they offer an accurate forecast. Are the forecast similar to what theory suggests?
I am also required to carry out the forecasting procedure mathematically for a few steps ahead. I assume I use the static forecast to carry this out?
I have attached a sample of the Eviews output for both my GJR model, dynamic and static forecast for reference.
Any help would be greatly appreciated.
"I am also required to carry out the forecasting procedure mathematically for a few steps ahead."
You simply need to take conditional expectation from both side of variance equation and derive the formula for h-step ahead variance forecasts.
We provided the formula in our recent paper on volatility forecasting. See the Appendix B Forecasting formulae on page 323-324.
A.S. Hasanov et al. (2018). Forecasting volatility in the biofuel feedstock markets in the presence of structural breaks: A comparison of alternative distribution functions. Energy Economics 70, 307–333.
Question
Using Eviews, how do I interpret the resulting coefficients in the conditional variance equation of this GJR-GARCH(1, 1)- MA(1) model?
I am required to write this model out by hand, however I am struggling in doing so.
I have attached a sample of the Eviews output for reference.
Volatility persistence in GJR model is given by
alpha + beta + gamma / 2 < 1
-0.014168 + 0.880880 + 0.149107/2 = 0.941266
In EViews output, RESID(-1) < 0 is an indicator (i.e., dummy) variable
(I_(t-1) = 1 if RESID(-1) < 0).
For a leverage effect, we look at gamma > 0 (0.149107 > 0).
Condition for non-negativity of variance equation coefficients:
c > 0
alpha > 0 (i.e., coefficient of RESID(-1)^2)
beta >= 0 (i.e., coefficient of GARCH(-1))
alpha + gamma >= 0.
That is, variance model is still valid even if alpha < 0 or gamma < 0 provided that alpha + gamma >= 0
Question
The conditional variance is specified to follow some latent stochastic process in some empirical applications of volatility modelling. Such models are referred to as stochastic volatility (SV) models which were originally proposed by Taylor (1986). The main issue in univariate SV model estimations is that the likelihood function is hard to evaluate because, unlike the estimation of GARCH family models, the maximum-likelihood technique has to deal with more than one stochastic error process. Nevertheless, recently, several new estimation methods such as quasi-maximum likelihood, Gibbs sampling, Bayesian Markov chain Monte Carlo, simulated maximum likelihood have been introduced for univariate models.
I would like to know whether any of aforementioned estimation methods have been extended to multivariate stochastic volatility models? Could anyone recommend any code, package or software with regard to the estimation of multivariate stochastic volatility models?
Following
Question
Could anyone suggest MATLAB toolbox or code for Markov switching volatility models for recursive or moving windows?
Thanks Peter for the links !!
Question
I do know that Eviews has an add on for this model, But I am using a old version of the Eviews and therefore the add on feature cannot be incorporated in the same.
The DCC model has become a standard model in many Econometric software: RATS, G@RCH under OxMetric, Microfit, MATLAB, rmgarch package in R.
Question
I basically want to know that whether this volatility modelling is the part of volatility analysis or both terms are completely different?
My above question on volatility is also in this context.....
Thanks and regards.....
Thanks..
Question
Dear all,
We know that while choosing any model for volatility we see our objective first like - To check spillover. To check news impact. To check leverage effect or clustering.
What exactly we are doing over here volatility analysis or volatility modelling?
Or can say that doing volatility analysis through volatility modelling as Egarch, bekk garch.
I mean whether these different aspect we are checking are covered under the word volatility analysis or the word volatility modelling?
Thanks and regards.....
Volatility analysis is done before modelling. Arch test, for instance, is part of the analysis which, if significant, provides ground for modelling.
Question
We are using GARCH model for checking the volatility of time series data. How can we check the Economic significance of the model ? Especially the extent to which the independent variable contribute to volatility of the model (ht) in each period.
ARCH and GARCH models have become important tools in the analysis of time series data, particularly in financial applications. These models are especially useful when the goal of the study is to analyze and forecast volatility. Attached article may help you to understand more about the economic significance of GARCH model.
Best Wishes,
Question
The JJ test for establishing a long run co integrating relationship and the EGARCH for verifying  short run dynamic linkages.
Would DCC-MGARCH be preferable to EGARCH and if so , why?
JJ test  only captures when there is a linear association between  sets data, however to assess linear and nonlinear association between sets of data, I advise using nonlinear cointegration test approach , like rank test  of Breitung.
best luck
Question
I am working on a volatility model for BDI indices and have encountered the following problems:
my garch(1;1) is non-stationary( coefficients in the garch term sum up to more than . I performed a sign and size bias test and discovered that size effects are significant, while sign effect is not. I tried estimating a TGARCH(1;1) and EGARCH(1;1) but in both of them the assymetry term is insignificant. Hence, I tried to steadily increase the order of EGARCH to (2:1) and (1:2), which did not help. Eventually I arrived to EGARCH(2;2) and added a risk premium term to mean equation. However, I have not had much experience with higher orders of garch models, so I have a couple of questions;
1) How do i interpret egarch(1;1) where all coeffcients are significant except the assymetry term?
2)What exactly do egarch(p,q) variables mean? I understand the importance of the assymetry term, but I cant really understand what the other variables in garch term account for?
3) I know the stationarity conditions for GARCH(1;1) process and stronger conditions of Bollershev, but how do I test for stationarity with higher orders of garch? Are there any ways to know it is stationary?
4) How do I choose the order of assymetry for egarch models of higher order?
5)Last but not least. As far as I understand my assymetry term is insignificant, which leads me to a question whethere there exist extensions of garch which account for SIZE effect, but do NOT account for SIGN effect?
Hi Carlos,
Thanks for you anser.
1) The data is stationary, so I would not say I have really extreme ouliers.
2)Yes, I am using Gaussian GARCH, I have also tried using GED distribution, but it appears to also be non-stationary.
3) Yes, of course I have specified the mean equation first.
Question
Testing for Volatility.
GARCH models are applied to returns, not prices. Returns are stationary, and therefore you do not need to worry about this issue.
Question
HI,
Can anyone share some insights about modeling volatility with explanatory variables except M_GARCH or GARCH family models ? I am particularly interested in developing a model that can better explain the volatility of international tourists flows with explanatory variables.
You might find my recent paper Pricing derivatives in a regime switching market with time inhomogeneous volatility relevant. link: http://arxiv.org/abs/1611.02026
In this model, the volatility is modeled as a particular type of pure jump process. The class of pure jump processes includes Markov chains. Although Markovian volatility is well studied, the analysis in a non-Markovian setup is not so common. In our paper, we have taken volatility as an independent semi-Markov process and analyzed the model.
Question
Based from the articles i read in modeling volatility, they computed returns of a certain variable, say price, as ln(p_t / p_(t-1)) where p_t is price at time t and p_(t-1) price at time t-1.However, in other studies, the returns is computed as (p_t - p_(t-1))/ p_(t-1). What is the difference between this two formula? is it right to just use the first formula instead of the second one? My variable is monthly interest rates, and i want to compute the returns for my analysis.
A good reference for a comparison of linear and log returns is the working paper of Attilio Meucci : "Quant Nugget 2: Linear vs. Compounded Returns – Common Pitfalls in Portfolio Management".
If the securities  used for computing returns are not too volatile and the time step of the return is short, the distribution of linear and compounded returns is similar but as the time step grows, so does the difference between the two distributions.
You decide whether to consider linear or compounded (log) returns based on your particular objective. In fact linear returns aggregate across assets thus risk and portfolio managers use linear returns for risk analysis, performance attribution, and portfolio optimization.  Compounded return aggregates across time and  thus log returns  are used for projections since it is easy to estimate their one period distribution and to project this distribution to a generic horizon into the future.
Question
I would liketo forcast the volatility compare among different volatility estimation models.
If your aim is to model the volatility of a series of futures contracts then you'll need to pay attention to the Samuelson effect and eventual seasonality in the volatility. You may take a look at these papers we wrote with Lorenz Schneider:
If your aim is to model a time series of a rolling proxy for a commodity's spot price (e.g. CL1 time series) then using a GARCH based econometrics model seems suitable. Depending on the chosen commodity you'll need a seasonal component (e.g. Natural Gas, Wheat)
Question
I run GARCH (1,1) to capture volatility spillover between spot and futures market. I find most of the coefficients are positive and some are negative. My doubt is why these coefficients are negative? Is there any justification on negative co-efficient? How to interpret and how to solve this problem? the assumption of GARCH model is that all co-efficients are positive. Can any one please clarify my doubt.  should I go for any other model?
Can you tell me how to check BEKK Framework in Eviews?
Question
Can any one help in modelling GARCH/EGARCH in Eviews or Stata?? I am stuck in modelling the multiple independent variables against single dependent one. Sample Results are attached for furtehr explanation. I am not sure , is it the right way or not?
Dear Malik,
I am including a PPT to explain how to model any GARCH type model in Eviews. I hope it will be beneficiary for you. Take care.
Mustafa Ozer
Question
While computing the Greeks in finance for stochastic volatility models, I ended up with a Skorohod integral. The problem is, how can i solve this integral numerically?
ISSUE: How to calculate the Skorohod integral?
STOCHASTIC PROCESS: Assume that there are random variables W(h) in Hilbert space H. The Hilbert space is a generalized Euclidean space, i.e. two dimension XY-coordinate system. The Hilbert H is comprised of elements h1, h2, … H. The random variable set W(h) in normally distributed in the Gaussian fashion (bell shaped with mean = 0) and there the mapping of h to W(h) is linear with mean:
(1)   E[W(h)] = 0
and variance:
(2)   E[W(g)W(h)]]
Assume now that there is a function u, and that u is given by:
(3)   U = sum(Fjhj)
… where Fj = smoothing function and hj element of H in Hilbert space.
The Skorohod integral (sigma) is given by:
(4)   Sigma(u) = sum(FjW(hj)) – [DFj, hj]H
… where DF = Malliavin derivative.
If you are working with financial data, you probably have the random variable set W(h) already, and the smoothing function Fj that accompanies it. I assume that the remaining issue is to find the Malliavin derivative for the second part of the equation: [DFj, hj]H. Since the first part sum(FjW(hj)) is a sum, you must have series of observations. Let's call this smoothed observations (FjW(hj)) as f1, f2, ... These are series of points once connected is a smoothed curve. in that same topology, there is a corresponding series of changes of the curve whose rate of change may be given by the derivative of the function that i associated with the curve; this part is the second portion of the equation: [DFj, hj]H. At the end, the Skohorod integral is simply the sum of all points defined by the function that provides the characteristics of the change minus the rate of change of that smoothed curve. Do not be confused by H and h from Hilbert space notation---it is line in space, i.e. just like XY space. H and h is used because we are dealing with special kind of stochastic process and the integration for that special stochastic process.
Question
Can anybody help me in getting this software? I have research work on MGARCH modelling.
You should contact Tom Doan at Estima Software.
I'm sure he could help.
Question
See above
Hi,
The answer lies in the relationship between the VIX index and the 30-day variance swap rate. Essentially, the squared VIX index approximates the conditional risk-neutral expectation of the annualized return variance, over the following 30 days, which is the 30-day variance swap rate.
The realized return variance can be split in different components showing that only OTM calls and puts give a contribution in its replication.
Please have a look at the attached paper (pp. 15-16), where an extensive explanation is provided.
Best regards
Question
I am interested in analyzing disparities in volatility in different five minute intervals in different trading days in the Forex market. Now I want to know whether the volatility in such intervals can be grouped into clusters and whether the clusters are the same for different trading days. Can anyone help me by suggesting a method or technique to achieve my objective of identifying clusters for volatility?
Why not taking the squared returns (or absolute returns) over the five minutes, take averages over days (for each five minute interval) and test the equality of means?
Question
Using the historical var-cov matrix as an input in the optimizer leads to estimation errors. What other methods can be used in estimating the var-covar apart from shrinkage and diagonal methods?
Implied Volatility as it is only the market's prediction of it from a Black- Scholes model, with quoted option prices , need not necessarily follow the statistical properties of a variance- co variance matrix .'Volatility smile' is an example for it
Question
Can anybody share this software? I have research using the MGARCH Model.
Maria, Refk and the great LUMENGO: Thanks for your suggestions
Question
Of course, ARIMA models are easily fitted in R. I estimate a model, in which the i-th error term of the ARIMA model shall depend on the three errors before (seasonal volatility). Is there any way to model seasonal volatility in R without using KFAS and/or similar packages? Does there exist any plug-in-method in any R-package, which is ready to use?
Looks like the free Ox version and G@rch package is available for version 6. Download link is at the end of chapter 1 http://www.timberlake.co.uk/slaurent/G@RCH/default.htm
Question
I am interested in studying the behavior of exchange rates. I would like to use R as a software. Can anyone please explain the algorithm that I should follow in order to fit a regime switch model? Do we need to know the number of regimes when fitting a RSM or is it possible for the model to identify the number of regimes?
EXCHANGE RATE & UNDERLYING REGIME
Exchange rate under a given regime would display very stable pattern. If the patterns change and that change is significant, it may be due to either (i) regime change, or (ii) economic shock. Regime change will cause the exchange rate to change as the market would adjust and readjust to the new regime. This new adjustment would become stabilized (see y(t) below). However, if it is a shock, the change would readjust itself to the old pattern because there is no institutional or fundamental change that would affect the exchange rate.
EXISTING REGIME
In order to detect regime change, one must have adequate data series over a period of time to allow the data to show its characteristic. The time series data would produce a characteristic pattern that confirms to the linear structure of:
(1) Y’ = a1 + bX + e
… where a1 = Y-intercept, b = independent factor, and e = forecast error. This model is a normal run of the existing regime.
In subsequent periods, b may vary, but the pattern of Y’ would still obey the same characteristic pattern. Assume then that there is a change:
(2) Y’’ = a2 + bX + e
And that a1 > a2 or a1 < a2. the Y-intercept of the two periods are not equal. This signifies that there has been a change that could be explained by the law of probability. It also follows the change of a1 to a2 was not also predictable. For this reason, it is necessary to construct a new model to accommodate a2.
REGIME SWITCHING
Generally, the abrupt change of the Y-intercept described above is not deterministic, i.e. could not have been reasonably predicted. Regime switching occurs in such a manner. While Y’ obeys a predictable patter with a fixed Y-intercept, the subsequent change produces a Y-intercept that is due to institutional change. Thus, Y’’ may be rewritten as:
(3) Y(regime) = C + bX + e
… where the Y-intercept C results from a two-state Markov chain: a period of a prior regime and a period under a new regime. The difference between (a1 – C) > 0 or (a1 – C) < 0; that is the difference is real and significant. The challenge is to locate the period when the switching occur.
DETECTION INDICATOR
The indicator used for regime switching is the Y-intercept in the equation Y = a + bX + c. Assume then that several samples had been taken, say several time periods, i.e. t1, t2, … tn, and that regression analysis is run for each period; the result shows that a1, a2, …, a^n are not equal. Does it mean that there is a regime switching? No. In order to constitute regime switching we need to use only two periods: one prior to regime switching and one after the regime switching. The challenge is to find this pair of Y-intercept at various time period.
One way to find the pair is to undertake series of data set over several time periods and list all their Y-intercepts for each period. A period that shows a peak or a trough from the mean is a suspected pair. The second step is to select the pair and verify whether there is a regime change. The verification may be accomplished by the Anderson-Darling test.
The Anderson-Darling test requires n > 5 or minimum sample size of 6. Here, the sample is the individual a1, a2, …, a^n or Y-intercepts for each period, i.e. 3 before and 3 after the suspected change. The critical value for the Anderson-Darling test A^2 at 0.95 confidence interval is 0.787; the decision rule follows: if A^2(observed) > 0.787, there has been a regime change, otherwise accept the null hypothesis (no regime change).
In order to prove that no Type I or Type II error had been committed, several segments of the data length of 6 to 7 elements (periods) may be retested. Prior periods where there is no regime change, the A^2 value would be less than 0.787. There will be a data segment that would break this pattern showing A^2 > 0.787.
NUMBER OF REGIMES
For purposes of constructing a predicting model, this issue may be moot. A system does not switch regime so often so as to necessitate the accounting for number of regimes used---unless one runs a data series that covers a span of many years or decades. For practical purposes, i.e. a specified segment of time series data, to proof the regime switching the number of regime is not relevant to the extent that it does not play a role in determining the current exchange rate. However, if the time series data prove that there had been several regimes switching in the period, then the switching frequency may be observed. However, this data (regime switching frequency) cannot be used to forecast potential switching in the new future---it only shows a past count.
GOODNESS-OF-FIT
After the evidence of regime has been verified then a model may be constructed. Use any conventional methods to test the goodness-of-fit and readjust the model accordingly. At this point, basic statistical tests combined with econometrics would be helpful.
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