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# 421 midterm 2 study guide Econ 421

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This 5 page Study Guide was uploaded by Theo Friedman on Sunday May 17, 2015. The Study Guide belongs to Econ 421 at University of Oregon taught by Caroline Weber in Spring 2015. Since its upload, it has received 186 views. For similar materials see Intro Econometrics in Economcs at University of Oregon.

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Date Created: 05/17/15

Midterm 2 Econ 421 Time series data 0 The time series for the X39s are at most weakly persistent Replaces drawn at random assumption 0 we replace i subscripts with t subscripts eg xi becomes xt and instead of n observations we now have T observations Static Model 0 modeling a contemporaneous relationship between y and the x39s 0 CtBOB1GDPtet Dynamic model 0 Models with lagged X39s 0 Ct BO B1 GDPt B2 GDPtl B3 GDPt392 et 0 Even if multicollinearity is present standard errors on the longrun effect are smaller 0 An Ftest will tell whether longrun effect is significant 0 Autoregressive Distributed Lag ADL models 0 ADLp q where p is the lags of the dependent variable used and q is the maximum number of lags of the X39s used Suppose we estimate as an ADL10 Ctb0blGDPtb2Ct1et This is implicitly allowing us to estimate a more complicated model with lags of GDP without including these lags directly In this model b1 captures the shortrun effect of GDP on C What about the equilibrium or longrun effect Cb0blGDPb2C 1b2Cb0blGDP Cb01b2b11b2GDP So the long run effect of GDP on C is b11b2 Bias Consistency and Asymptotic Normality For OLS to be unbiased we must have EutX 0 which we can split into two parts 1 Disturbance is independent of X39s in same observation 2 Disturbance is independent of X39s in other observation Ebl EB1 Etxtxutxtxquot2 B1 B Consistency prove consistency for the following simple regression modeL Ctb0b1Ct1u So don39t need 2 to hold just need 1 Eutjxt 0 to hold Asymptotic Normality and ttests When we have consistency we also have asymptotic normality Obtaining the asymptotic distribution can take a while so ttests may not be as illuminating as we would like in nite samples Worst when Bl close to one and had to impose that Bllt1 Autocorrelation covutus 0 when ts Violates key assumption 1 positive autocorrelation 2 negative correlation Static AR1 ytBOletut utput1et AR2 would be utp1ut1p2ut2et Just like for heteroskedasticity with autocorrelation in a static model or a dynamic model with lagged X39s coefficients remain unbiased but standard errors do not still inefficient Tests to detect autocorrelation Static Model or a dynamic model with lagged X39s Suppose we have the following static model with an AR1 error term ytBOB1xtu utptut1et 1 Estimate the model ytb0b1xtet 2 Calculate the residual set 3 Specify the AR process we want to test and estimate this regression etpet1t no constant in this regression 4Conduct a ttest where the null hypothesis is p 0 If we reject the null hypothesis we reject no autocorrelation NOTE Could also perform a BreuschGodfrey test using the same setup Dynamic Model with lagged 1 Estimate the model ytb0b1xtb2yt1 et 2 Calculate the residua lset 3 Specify the AR process we want to test and estimate this regression suppose AR2 eta0a1xta2yt1p1et1p1et2t13 NOTE Including a constant xt and yt1 is now very important because the residual is inconsistent if there is autocorrelation UJN 3 4 Conduct an LM test where the null hypothesis is p1p2 O This test is specified as follows nRA2 xquot2p where n is the number of observations in the regression from step 3 and RA2 is the Rquot2from the regression in step 3 If we reject the null hypothesis we reject that et is not auto correlated Fixes for the static model or dvnamic model with laooed X39s the static model produces consistent but inef cient estimates and inconsistent standard errors A Examine whether model is misspeciffed An omitted variable will generate the appearance of autocorrelation over time if it is correlated over time B Serial correlation robust NeweyWest standard errors Intuition is the same as for heteroskedasticityrobust standard errors GLSFGLS Suppose we have the following static model with an AR1 error term ytBOletut Where utput1et Don39t know p So we implement FGLS The steps are Estimate the model ytb0b1xtet18 Calculate the residuals et Estimate the regression to obtain pthe sample estimate of p etpet1t no constant in this regression Now estimate ytb01p b1xtpxt1 pyt1vt Still consistent but not unbiased NonStationary Time Series Weak Dependence Weak dependence as h goes to in nity xt and xth become almost independent Need weak dependence for OLS We are going to operationalize this notion with some form of stationarity for now covariance stationarity Conditions necessary for covariance stationarity 1 The mean of the distribution is independent of time ieExt Ext1 forall t 2 The variance of the distribution is independent of timevarxt varxt1ampquot2 for all t The covariance between xt and any x th depends only on h not on t Random Walk xtxt1et1 Famousinfamous example of randomwalk stock market prices show graph of random walk over time one up one down There are several different ways to test EMH taking rst differences of both sides of Change xt Changextl Change et estimate Change xtBO Bl Change xt1 Change et Run a ttest where the null is Bl 1 if BO 0 this means random walk without drift Time trends What if the data has a time trend Suppose the true model is given by yt01xt2tut Not covariance stationary With time trend included it is trendstationary Can make covariance stationary by taking rst differences Panel Data Treatment Effect the effect of an increase in the minimum wage on employment in New Jersey Card and Krueger 1994 Call the estimate of the effect of the polioyatreatment effect Fixed Effects Y itBOB1MlNWAGE it 1 NJ it 1BZ MINWAGE it 1 BSNJ it 1 u it First Differences with two years of data Change Y itB1NJ it 1 BZ Change u it with two years of data both methods are equivalent With more they are not Fixed Effects FE and Firstdifferences FD Think of fixed effects as including an individualspecific intercept Assumptions for OLS regression with fixed effects A A mathematical representation of the process generating the data that is linear in the parameters with an additive disturbance or if it is not linear in the parameters it can be made linear for example taking logs of both sides to linearize it Model is correctly specified B Properties of variables 0 The time series for the X39s are at most weakly persistent and the crosssection is drawn at random X39s have variation over time and across individuals at a given time X39s are exogenousEuithiai O X39s are measured without error measurement a b deal in this context X39s causes Y but Y does not cause X39s The X39s are not linearly dependent C Properties of disturbance Eu it 0 zero mean Eu2 it 2i 2 homoskedastic variance Euiit uil Eut Eul 0 no correlation in the error terms across time Unlikely to hold Comparing FE and FD if no serial correlation in the error term we know FE is BLUE so that is the 0 Residual follows a random walk FD is much better as it eliminates the autocorrelation Normally it is difficult to know which is more efficient When the time span is long the assumption that i39s FE isn t timevarying becomes an untenable assumption so first differences are preferred Random Effect Models We can just pool all the data over time and not estimate the FE lf random effects is can identify the effect of a variable that does not change with time but if we are really interested in this feature we could use Correlated Random Effects Wages it b0b1unionitb2unionie it b2 is capturing the effect of any permanent skill differences between those unionized and those no

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