models in his dissertation, giving the method its name. Wright attempted to determine the supply and demand for butter using panel data on prices and quantities sold in the United States. The idea was that a regression analysis could produce a demand or supply curve because they are formed by the path between prices and quantities demanded or supplied. The problem was that the observational data did not form a demand or supply curve as such, but rather a cloud of point observations that took different shapes under varying market conditions. It seemed that making deductions from the data remained elusive. The problem was that price affected both supply and demand so that a function describing only one of the two could not be constructed directly from the observational data. Wright correctly concluded that he needed a variable that correlated with either demand or supply but not both – that is, an instrumental variable. After much deliberation, Wright decided to use regional rainfall as his instrumental variable: he concluded that rainfall affected grass production and hence milk production and ultimately butter supply, but not butter demand. In this way he was able to construct a regression equation with only the instrumental variable of price and supply. Theory While the ideas behind IV extend to a broad class of models, a very common context for IV is in linear regression. Traditionally, an instrumental variable is defined as a variable Z that is correlated with the independent variable X and uncorrelated with the "error term" U in the linear equation is a vector. is a matrix, usually with a column of ones and perhaps with additional columns for other covariates. Consider how an instrument allows to be recovered. Recall that OLS solves for such that (when we minimize the sum of squared errors, , the first-order condition is exactly .) If the true model is believed to have due to any of the reasons listed above—for example, if there is an omitted variable which affects both and separately—then this OLS procedure will not yield the causal impact of on . OLS will simply pick the parameter that makes the resulting errors appear uncorrelated with . Consider for simplicity the single-variable case. Suppose we are considering a regression with one variable and a constant (perhaps no other covariates are necessary, or perhaps we have partialed out any other relevant covariates): In this case,