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Lecture 8

Lecture 8 - Introduction to Econometrics Lecture 8...

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Introduction to Econometrics 8-1 Technical Notes Introduction to Econometrics Lecture 8: Forecasts and Structural Breaks Forecasting and Forecast Confidence Intervals In a simple regression, if we know the forecast value of the regressor X p , we can forecast Y conditional on X using Y p = α e + β e X p since E[u p ] = 0. This forecast should be contrasted with the unconditional forecast of Y given by Y u = μ e where μ e is the estimated sample mean of Y 1 . The forecast error e p is e p = Y - Y p = ( α + β X p + u p ) - ( α e + β e X p ) = α - α e + ( β - β e )X p + u p with expectation E[e p ] = {E[ α - α e ] + X p E[ β - β e ]} + E[u p ] = 0 because the OLS estimates of the parameters are unbiased. Therefore the forecast is unbiased, but forecast error arises (even if we know X p ) both because we cannot forecast the random variable u p , and also because our estimates of α and β are subject to sampling error. This sampling error component (the term in braces) depends on the sample disturbances, but not on u p , so the two types of error are independent. The two components of the forecasting error are both Normally distributed, so their sum is also Normally distributed: this result can be generalised easily to multiple regression. It is also possible to show that forecasts based on OLS regression coefficients have lower variance than other linear unbiased forecasts: this is an analogue of the Gauss-Markov theorem that the OLS parameter estimators are BLUE. Using Forecasts - Prediction There are two main ways of using these results on the distribution of a forecast. The first of these corresponds to the conventional use of the word ‘forecast’: an attempt to predict the outcome for Y for an observation where we know (or can predict) the explanatory variables. Although very often the ‘users’ of such predictions are only interested in the single central value Y p , it is also important to provide a measure of forecast precision in the form of a confidence interval. To compute the forecast confidence interval for a simple regression, compute Var[e p ] = E[{e p - E[e p ]} 2 ] = E[{( α - α e ) + ( β - β e )X p + u p } 2 ] = {Var( α e ) + Var( β e )(X p ) 2 + 2Cov( α e , β e )X p } + Var(u p ) 1 In this discussion of forecasts we concentrate on the case where the data is generated by random sampling, so forecasting amounts to predicting the value of Y for another draw from the underlying population. The problems involved in forecasting the value of a time-series variable are discussed later in the course. However it is helpful to note here that for time series the distinction between unconditional and conditional forecasts is less useful than that between forecasts which use only the past history of Y (‘univariate forecasts’) and those which use other variables (‘multivariate forecasts’).
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Introduction to Econometrics 8-2 Technical Notes since E[( α - α e )u p ] = E[( β - β e )X p u p ] = 0, because u p is independent of the sample
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Lecture 8 - Introduction to Econometrics Lecture 8...

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