Econ 423 Lecture Notes

Transcription

1 Econ 423 Lecture Notes (hese notes are modified versions of lecture notes provided by Stock and Watson, hey are for instructional purposes only and are not to be distributed outside of the classroom.) 5-

2 Heteroskedasticity and Autocorrelation-Consistent (HAC) Standard Errors Consider a generalization of the distributed lag model, where the errors u t are not necessarily i.i.d., i.e., Y t = β 0 + β X t + + β r+ X t r + u t. Suppose that u t is serially correlated; then, OLS will still yield consistent* estimators of the coefficients β 0, β,., β r+ (*consistent but possibly biased!) he sampling distribution of ˆ β, etc., is normal 5-2

3 BU the formula for the variance of this sampling distribution is not the usual one from cross-sectional (i.i.d.) data, because u t is not i.i.d. in this case since, in particular, u t is serially correlated! his means that the usual OLS standard errors (usual SAA printout) are wrong! We need to use, instead, SEs that are robust to autocorrelation as well as to heteroskedasticity his is easy to do using SAA and most (but not all) other statistical software. 5-3

4 HAC standard errors, ctd. he math for the simplest case with no lags: Y t = β 0 + β X t + u t he OLS estimator: Using the usual regression algebra, we obtain ( X t X ) u ˆ t= β β = 2 ( X t X ) t= t vt t= 2 σ X where v t = (X t X )u t. (in large samples) 5-4

5 HAC standard errors, ctd. hus, in large samples, var( ˆ β ) = var = 2 t = s = vt / t = ( σ ) 2 2 X cov( vt, vs )/( σ X ) 2 2 In i.i.d. cross sectional data, cov(v t, v s ) = 0 for t s, so var( ˆ β ) = var( v ) 2 t )/ = t ( σ ) = 2 2 X σ ( σ ) 2 v 2 2 x his is our usual cross-sectional result. 5-5

6 HAC standard errors, ctd. But in time series data, cov(v t, v s ) 0 in general. Consider = 2: var vt = var[½(v +v 2 )] t = = ¼[var(v ) + var(v 2 ) + 2cov(v,v 2 )] = ½ 2 2 σ v + ½ρ σ (ρ = corr(v,v 2 )) = ½σ 2 v f 2, where f 2 = (+ρ ) In i.i.d. data, ρ = 0 so f 2 =, yielding the usual formula In time series data, if ρ 0 then var( ˆ β ) is not given by the usual formula. v 5-6

7 Expression for var( ˆ β ), general var so var( ˆ β ) = where f = vt = t = 2 σ v f σ ( σ ) 2 v 2 2 X f j + 2 ρ j j= Conventional OLS SE s are wrong when u t is serially correlated (SAA printout is wrong). he OLS SEs are off by the factor f We need to use a different SE formula!!! 5-7

8 HAC Standard Errors Conventional OLS SEs (heteroskedasticity-robust or not) are wrong when u t is autocorrelated So, we need a new formula that produces SEs that are robust to autocorrelation as well as heteroskedasticity We need Heteroskedasticity- and Autocorrelation- Consistent (HAC) standard errors If we knew the factor f, we could just make the adjustment. However, in most practical applications, we must estimate f. 5-8

9 HAC SEs, ctd. 2 var( ˆ σ v j β ) = 2 2 ( σ X ) f, where f = + 2 ρ j j= he most commonly used estimator of f is: f ˆ = ρ% j is an estimator of ρ j m m j + 2 % j j m ρ (Newey-West) = his is the Newey-West HAC SE estimator m is called the truncation parameter Why not just set m =? hen how should you choose m? o Use the Goldilocks method o Or, use the rule of thumb, m = 0.75 /3 5-9

10 Empirical Example he Orange Juice Data Data Monthly, Jan. 950 Dec ( = 62) Price = price of frozen OJ (a sub-component of the producer price index; US Bureau of Labor Statistics) %ChgP = percentage change in price at an annual rate, so %ChgP t = 200 ln(price t ) FDD = number of freezing degree-days during the month, recorded in Orlando FL o Example: If November has 2 days with lows < 32 o, one at 30 o and at 25 o, then FDD Nov = = 9 5-0

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12 Initial OJ regression % ChgP t = FDD t (.22) (.3) Statistically significant positive relation More freezing degree days price increase Standard errors are heteroskedasticity and autocorrelationconsistent (HAC) SE s 5-2

13 Example: OJ and HAC estimators in SAA. gen l0fdd = fdd; generate lag #0. gen lfdd = L.fdd; generate lag #. gen l2fdd = L2.fdd; generate lag #2. gen l3fdd = L3.fdd;.. gen l4fdd = L4.fdd;.. gen l5fdd = L5.fdd;.. gen l6fdd = L6.fdd;. reg dlpoj fdd if tin(950m,2000m2), r; NO HAC SEs Linear regression Number of obs = 62 F(, 60) = 2.2 Prob > F = R-squared = Root MSE = Robust dlpoj Coef. Std. Err. t P> t [95% Conf. Interval] fdd _cons

14 Example: OJ and HAC estimators in SAA, ctd Rerun this regression, but with Newey-West SEs:. newey dlpoj fdd if tin(950m,2000m2), lag(7); Regression with Newey-West standard errors Number of obs = 62 maximum lag: 7 F(, 60) = 2.23 Prob > F = Newey-West dlpoj Coef. Std. Err. t P> t [95% Conf. Interval] fdd _cons Uses autocorrelations up to m = 7 to compute the SEs rule-of-thumb: 0.75*(62 /3 ) = 6.4 7, rounded up a little. OK, in this case the difference in SEs is small, but not always so! 5-4

15 Example: OJ and HAC estimators in SAA, ctd.. global lfdd6 "fdd lfdd l2fdd l3fdd l4fdd l5fdd l6fdd";. newey dlpoj $lfdd6 if tin(950m,2000m2), lag(7); Regression with Newey-West standard errors Number of obs = 62 maximum lag : 7 F( 7, 604) = 3.56 Prob > F = Newey-West dlpoj Coef. Std. Err. t P> t [95% Conf. Interval] fdd lfdd l2fdd l3fdd l4fdd l5fdd l6fdd _cons global lfdd6 defines a string which is all the additional lags What are the estimated dynamic multipliers (dynamic effects)? 5-5

16 FAQ: Do I need to use HAC SEs when I estimate an AR or an ADL model? A: No, only if one is sure that the true model is an AR or an ADL in the purest sense so that there is no serial correlation or heteroskedasticity in the errors. In AR and ADL models with homoskedastic errors, one may argue that the errors will be serially uncorrelated if you include enough lags of Y o If you include enough lags of Y, then the error term can t be predicted using past Y, or equivalently by past u so u is serially uncorrelated However, the safer and more robust choice would be to always use HAC SE s. 5-6