Fall 2008 under econometrics prof keunkwan ryu 9

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Fall 2008 under Econometrics Prof. Keunkwan Ryu 9 Variances of OLS Estimators Just as in the cross-section case, we need to add an assumption of homoskedasticity in order to be able to derive variances Now we assume Var( u t | X ) = Var( u t ) = σ 2 Thus, the error variance is independent of all the x ’s, and it is constant over time We also need the assumption of no serial correlation: Corr( u t ,u s | X )=0 for t s
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 10 OLS Variances (continued) Under these 5 assumptions, the OLS variances in the time-series case are the same as in the cross-section case. Also, The estimator of σ 2 is the same OLS remains BLUE With the additional assumption of normal errors, inference is the same
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 11 Trending Time Series Economic time series often have a trend Just because 2 series are trending together, we can’t assume that the relation is causal Often, both will be trending because of other unobserved factors Even if those factors are unobserved, we can control for them by directly controlling for the trend
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 12 Trends (continued) One possibility is a linear trend, which can be modeled as y t = α 0 + α 1 t + e t , t = 1, 2, … Another possibility is an exponential trend, which can be modeled as log( y t ) = α 0 + α 1 t + e t , t = 1, 2, … Another possibility is a quadratic trend, which can be modeled as y t = α 0 + α 1 t + α 2 t 2 + e t , t = 1, 2, …
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