Time series data y,=Bo+B Brit ◆1. Basic analysis Economics 20- Prof anderson
Economics 20 - Prof. Anderson 1 Time Series Data yt = b0 + b1 xt1 + . . .+ bk xtk + ut 1. Basic Analysis
Time series vs. Cross sectional Time series data has a ter mporal ordering, unlike cross-section data o Will need to alter some of our assumptions to take into account that we no longer have a random sample of individuals Instead we have one realization of a stochastic (i.e. random) process Economics 20- Prof anderson
Economics 20 - Prof. Anderson 2 Time Series vs. Cross Sectional Time series data has a temporal ordering, unlike cross-section data Will need to alter some of our assumptions to take into account that we no longer have a random sample of individuals Instead, we have one realization of a stochastic (i.e. random) process
Examples of Time Series models o A static model relates contemporaneous variables: y,= Bo+ B=+ o A finite distributed lag (FDL) model allows one or more variables to affect y with a lag y=0+C=1+8+82+l o More generally, a finite distributed lag model of order g will include g lags of z Economics 20- Prof anderson
Economics 20 - Prof. Anderson 3 Examples of Time Series Models A static model relates contemporaneous variables: yt = b0 + b1 zt + ut A finite distributed lag (FDL) model allows one or more variables to affect y with a lag: yt = a0 + d0 zt + d1 zt-1 + d2 zt-2 + ut More generally, a finite distributed lag model of order q will include q lags of z
Finite Distributed lag models ◆ We can cal all So the impact propensity -it reflects the immediate change in y For a temporary. 1-period change to its original level in period q+7 y returns ◆ We can call S+,+…+ the long-run propensity (lrp)it reflects the long-run change in y atter a permanent change Economics 20- Prof anderson 4
Economics 20 - Prof. Anderson 4 Finite Distributed Lag Models We can call d0 the impact propensity – it reflects the immediate change in y For a temporary, 1-period change, y returns to its original level in period q+1 We can call d0 + d1 +…+ dq the long-run propensity (LRP) – it reflects the long-run change in y after a permanent change
Assumptions for unbiasedness o Still assume a model that is linear in parameters:y-Bo+ Bx+...+ Bkxuk+ Still need to make a zero conditional mean assumption: E(uX=0, t=1, 2,...,n e Note that this implies the error term in any given period is uncorrelated with the explanatory variables in all time periods Economics 20- Prof anderson 5
Economics 20 - Prof. Anderson 5 Assumptions for Unbiasedness Still assume a model that is linear in parameters: yt = b0 + b1 xt1 + . . .+ bk xtk + ut Still need to make a zero conditional mean assumption: E(ut |X) = 0, t = 1, 2, …, n Note that this implies the error term in any given period is uncorrelated with the explanatory variables in all time periods
Assumptions(continued) This zero conditional mean assumption implies the x's are strictly exogenous o An alternative assumption, more parallel to the cross-sectional case, Is E(ulx=0 e This assumption would imply the x's are contemporaneously exogenous o Contemporaneous exogeneity will only be sufficient in large samples Economics 20- Prof anderson 6
Economics 20 - Prof. Anderson 6 Assumptions (continued) This zero conditional mean assumption implies the x’s are strictly exogenous An alternative assumption, more parallel to the cross-sectional case, is E(ut |xt ) = 0 This assumption would imply the x’s are contemporaneously exogenous Contemporaneous exogeneity will only be sufficient in large samples
Assumptions(continued) Still need to assume that no x is constant and that there is no perfect collinearity e Note we have skipped the assumption of a random sample e The key impact of the random sample assumption is that each u: is independent Our strict exogeneity assumption takes care of it in this case Economics 20- Prof anderson 7
Economics 20 - Prof. Anderson 7 Assumptions (continued) Still need to assume that no x is constant, and that there is no perfect collinearity Note we have skipped the assumption of a random sample The key impact of the random sample assumption is that each ui is independent Our strict exogeneity assumption takes care of it in this case
Unbiasedness of ols o Based on these 3 assumptions, when using time-series data, the ols estimators are unbiased e Thus, just as was the case with cross section data, under the appropriate conditions ols is unbiased e Omitted variable bias can be analyzed in the same manner as in the cross-section case Economics 20- Prof anderson 8
Economics 20 - Prof. Anderson 8 Unbiasedness of OLS Based on these 3 assumptions, when using time-series data, the OLS estimators are unbiased Thus, just as was the case with crosssection data, under the appropriate conditions OLS is unbiased Omitted variable bias can be analyzed in the same manner as in the cross-section case
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 o Now we assume var(u X)=var(u=0 o 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 ws 1X)=0fort≠S Economics 20- Prof anderson 9
Economics 20 - Prof. Anderson 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(ut |X) = Var(ut ) = s 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(ut ,us | X)=0 for t s
OLS Variances(continued) o 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 o2 is the same ◆ oLS remainS BLue e With the additional assumption of normal errors. inference is the same Economics 20- Prof anderson 10
Economics 20 - Prof. Anderson 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 s 2 is the same OLS remains BLUE With the additional assumption of normal errors, inference is the same
