Lecture 8

# Lecture 8 - Forecasting and regression For a simple...

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1 Forecasting and regression For a simple regression, given • OLS estimates of the regression coefficients α e , β e • the forecast value of X, X p The point forecast is y p = α e + β e X p since E[u p ] = 0 The forecast error 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 The error arises from uncertainty about u p sampling errors in α e , β e The variance of the forecast The expected forecast error is E[e p ] = E[( α - α e ) + ( β - β e )X p + u p ] = 0 α e , β e are unbiased and E[u p ] = 0 Hence the forecast is unbiased, and Variance = MSE The forecast variance is Var[e p ] = {Var[ α e ] + Var[ β e ](X p ) 2 + 2Cov[ α e , β e ]X p } + Var[u p ] = σ 2 [{(1/N)(1 + (X p -M(X)) 2 /Var N (X))}+1] First component (in {}) is uncertainty of the estimated model: function of sample disturbances Second component is ‘true’ uncertainty about disturbance for forecast observation Random sampling ensures that these two components are independent Forecast confidence intervals The forecast is a linear function of the regression coefficients and u p Hence if coefficients and u p are both Normal so is the forecast error The 95% confidence interval (with σ 2 unknown) for the forecast is y p -t N-2 SE[e p ] < y < y p +t N-2 SE[e p ] replacing the Normal distribution by the t-distribution Forecasts based on multiple regressions are also unbiased and normally distributed, but the variance formula is more complex

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2 An example of forecasting: the wage of a male worker We forecast the wage of a male worker with 12 years of schooling and 6 years of work experience An unconditional
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Lecture 8 - Forecasting and regression For a simple...

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