1 / 43

Time Series Data

Time Series Data. y t = b 0 + b 1 x t1 + . . .+ b k x tk + u t Basic Analysis. Time Series vs. Cross Sectional. Time series data has a temporal ordering, unlike cross-section data

jovan
Download Presentation

Time Series Data

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Time Series Data yt = b0 + b1xt1 + . . .+ bkxtk + ut Basic Analysis Slides by K. Clark based on P.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

  3. Examples of Time Series Models • A static model relates contemporaneous variables: yt = b0 + b1zt + ut • A finite distributed lag (FDL) model allows one or more variables to affect y with a lag: • yt = a0 + d0zt + d1zt-1 + d2zt-2 + ut • More generally, a finite distributed lag model of order q will include q lags of z

  4. Lagged Dependent Variable Models • Another common type of time series model is where one or more lags of the dependent variable appear, e.g. • yt = a0 + d0yt-1 + d1yt-2 + d2yt-3 + ut • Such models are not considered in ES5611 but reappear in ES5622 • Here they are ruled out by assumption TS.2

  5. Assumptions for Unbiasedness • [TS.1] Assume a model that is linear in parameters: yt = b0 + b1xt1 + . . .+ bkxtk + ut • [TS.2] 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 • This assumption also called strict exogeneity

  6. Assumptions (continued) • 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

  7. Assumptions (continued) • [TS.3] 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

  8. Unbiasedness of OLS • Based on these 3 assumptions, when using time-series data, the OLS estimators are unbiased • Omitted variable bias can be analyzed in the same manner as in the cross-section case

  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 • [TS.4] Assume Var(ut|X) = Var(ut) = s2 • Thus, the error variance is independent of all the x’s, and it is constant over time • [TS.5] Assume no serial correlation: Cov(ut,us| X) = 0 for t  s

  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 s2 is the same • OLS remains BLUE (Gauss-Markov) • With the additional assumption of normal errors, inference is the same

  11. Example using Microfit (10.3) • Microfit 4 available on all networked computers • Econometric package geared towards time series analysis • Mainly menu driven package • Has some quirks • Practice exercise and handout on Web page but not expecting you to become proficient - more in ES5622

  12. Example using Microfit (cont’d) • Castillo-Freeman and Freeman (1992): effect of minimum wage on employment in Puerto Rico • Variables: prepop - employment rate, mincov - ‘importance’ of minimum wage, • Simple Model: • log(prepop) = b0 + b1log(mincov) + u • Clear prediction from economic theory of sign of b1

  13. Microfit output - page 1 Ordinary Least Squares Estimation ******************************************************************************* Dependent variable is LPREPOP 38 observations used for estimation from 1950 to 1987 ******************************************************************************* Regressor Coefficient Standard Error T-Ratio[Prob] CONSTANT -1.1598 .027281 -42.5120[.000] LMINCOV -.16296 .019481 -8.3650[.000] ******************************************************************************* R-Squared .66029 R-Bar-Squared .65085 S.E. of Regression .054939 F-stat. F( 1, 36) 69.9728[.000] Mean of Dependent Variable -.94407 S.D. of Dependent Variable .092978 Residual Sum of Squares .10866 Equation Log-likelihood 57.3657 Akaike Info. Criterion 55.3657 Schwarz Bayesian Criterion 53.7282 DW-statistic .34147 ******************************************************************************* • Note where all the usual stuff is

  14. Microfit output - page 2 Diagnostic Tests ******************************************************************************* * Test Statistics * LM Version * F Version * ******************************************************************************* * * * * * A:Serial Correlation*CHSQ( 1)= 25.8741[.000]*F( 1, 35)= 74.6828[.000]* * * * * * B:Functional Form *CHSQ( 1)= 2.8662[.090]*F( 1, 35)= 2.8553[.100]* * * * * * C:Normality *CHSQ( 2)= .071873[.965]* Not applicable * * * * * * D:Heteroscedasticity*CHSQ( 1)= 4.3213[.038]*F( 1, 36)= 4.6192[.038]* ******************************************************************************* A:Lagrange multiplier test of residual serial correlation B:Ramsey's RESET test using the square of the fitted values C:Based on a test of skewness and kurtosis of residuals D:Based on the regression of squared residuals on squared fitted values

  15. 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 - leads to spurious regression • Even if those factors are unobserved, we can control for them by directly controlling for the trend

  16. Example - trending data • UK aggregate consumption and income 1948-85, annual data • Note extremely high correlation in scatter plot • R2 = 0.9974

  17. Trends (continued) • One possibility is a linear trend, which can be modeled as yt = a0 + a1t + et, t = 1, 2, … • Another possibility is an exponential trend, which can be modeled as log(yt) = a0 + a1t + et, t = 1, 2, … • Another possibility is a quadratic trend, which can be modeled as yt = a0 + a1t + a2t2 + et, t = 1, 2, …

  18. Detrending • Adding a linear trend term to a regression is the same thing as using “detrended” series in a regression • Detrending a series involves regressing each variable in the model on t • The residuals form the detrended series • Basically, the trend has been partialled out

  19. Detrending (continued) • An advantage to actually detrending the data (vs. adding a trend) involves the calculation of goodness of fit • Time-series regressions tend to have very high R2, as the trend is well explained • The R2 from a regression on detrended data better reflects how well the xt’s explain yt

  20. Example again • Define time trend variable, t • Original model: Model with trend:

  21. Example (cont’d) • Scatter plot of detrended series • R2 = 0.7868 • Still high but a more accurate measure of how well Y explains C • However, these data may be highly persistent and simple methods not appropriate (Wooldridge, Ch.11, ES5622)

  22. Time Series Data yt = b0 + b1xt1 + . . .+ bkxtk + ut Serial Correlation Slides by K. Clark based on P.Anderson

  23. Serial Correlation defined • Serial correlation (autocorrelation) is where TS.5 does not hold • Cov(ut,us¦X)  0 for t  s. • A particular form of serial correlation is extremely common in time series data • This is because ‘shocks’ tend to persist through time

  24. Implications of Serial Correlation • Unbiasedness (or consistency) of OLS does not depend on TS.5 • However OLS is no longer BLUE when serial correlation is present • And OLS variances and standard errors are biased • Hence usual inference procedures are not valid

  25. The AR(1) Process • First order autoregressive error process a useful model of serial correlation • yt = b0 + b1xt1 + . . .+ bkxtk + ut • ut = rut-1 + et |ρ| < 1 • where et are uncorrelated random variables with mean 0 and variance σe2 • Typically expect r > 0in economic data • Positive serial correlation

  26. The AR(1) Process • E(ut) = 0 • Var(ut) = σe2/(1-r2) • Cov(ut, ut+j) = rjVar(ut) • Corr(ut, ut+j) = rj • So the error term is zero mean, homoscedastic but has serial correlation which is positive if ρ > 0

  27. Estimator variance: simple regression This is not equal to the usual formula since Cov(ut, us)  0 in the presence of serial correlation

  28. Testing for AR(1) Serial Correlation • Want to be able to test for whether the errors are serially correlated or not • Want to test H0: r = 0 in ut = rut-1 + et, t =2,…, n, where ut is the regression model error term • With strictly exogenous regressors, an asymptotically valid test is very straightforward – simply regress the residuals on lagged residuals and use a t-test

  29. Testing for AR(1) Serial Correlation (continued) • An alternative is the Durbin-Watson (DW) statistic, which is calculated by many packages • If the DW statistic is around 2, then we can reject serial correlation, while if it is significantly < 2 we cannot reject • Critical values are in the form of ‘bounds’: reject if DW<dL, do not reject if DW>dU, inconclusive otherwise. Tables available.

  30. Testing for AR(1) Serial Correlation (continued) • Note that the t-test is only valid asymptotically while DW has an exact distribution under classical assumptions including normality. • If the regressors are not strictly exogenous, then neither the t nor DW test are valid • Instead regress the residuals (or y) on the lagged residuals and all of the x’s and use a t-test

  31. Testing for Higher Order S.C. • Can test for AR(q) serial correlation in the same basic manner as AR(1) • Just include q lags of the residuals in the regression and test for joint significance • Can use F test or LM test, where the LM version is called a Breusch-Godfrey test and is (n-q)R2 using R2 from auxiliary (residual) regression

  32. Example • In the Puerto Rico minimum wage example, DW = 0.3415 • dL = 1.535 so we can reject H0: r = 0 against H1: r > 0 • Assuming strict exogeneity, an auxiliary regression gives: • Hence reject H0 • S/C is present

  33. Correcting for Serial Correlation • Start with case of strictly exogenous regressors, and maintain all G-M assumptions except no serial correlation • Assume errors follow AR(1) so • ut = rut-1 + et, t =2,…, n • Var(ut) = s2e/(1-r2) • We need to try and transform the equation so we have no serial correlation in the errors

  34. Correcting for S.C. (continued) • Use a simple regression model for convenience: • Consider that since yt = b0 + b1xt + ut , then • yt-1 = b0 + b1xt-1 + ut-1 • If you multiply the second equation by r, and subtract if from the first you get • yt– r yt-1 = (1– r)b0 + b1(xt – r xt-1) + et , since et= ut– r ut-1 • This quasi-differencing results in a model without serial correlation

  35. Feasible GLS Estimation • OLS applied to the transformed model is GLS and is BLUE • Problem: don’t know r, so we need to get an estimate first • Can use estimate obtained from regressing residuals on lagged residuals without an intercept • This procedure is called Cochrane-Orcutt estimation

  36. Feasible GLS Estimation • One slight problem with Cochrane-Orcutt is that we lose an observation (t = 1) • The Prais-Winsten method corrects this by multiplying the first observation by (1-ρ2)1/2 and including it in the model • Asymptotically the two methods are equivalent • But in small sample, time-series applications two methods can get different answers

  37. Feasible GLS (continued) • Often both Cochrane-Orcutt and Prais-Winsten are implemented iteratively • This basic method can be extended to allow for higher order serial correlation, AR(q) • Most statistical packages including Microfit will automatically allow for the estimation of such models without having to do the quasi-differencing by hand

  38. FGLS versus OLS • In the presence of serial correlation OLS is unbiased, consistent but inefficient • FGLS is consistent and more efficient than OLS if serial correlation is present and the regressors are strictly exogenous • However OLS is consistent under a weaker set of assumptions than FGLS • So choice of estimator depends on weighing up different criteria

  39. Serial Correlation-Robust Standard Errors • It’s possible to calculate serial correlation-robust standard errors, along the same lines as heteroskedasticity robust standard errors • Idea is to scale the OLS standard errors to take into account serial correlation • The details of this are beyond our scope - the method is implemented in Microfit where it goes by the name of Newey-West

  40. Example • In Puerto Rico minimum wage example: • found serial correlation • compute Cochrane-Orcutt estimates by hand • find iterated C-O estimates • Find Newey-West s/c robust standard errors • Demonstrate in Microfit • Results compared on next slide

  41. Example: summary of results

  42. Next Week and beyond • Next Week (8/12/03): Go through mock exam answers in usual lecture time and place • KC will keep office hours (Wed 10-12) while the University is open • Email for appointment outside these times (ken.clark@man.ac.uk) • Check with tutors for their availability

  43. Next Semester - ES5622 • 5 lectures on Cross Section econometrics by Dr Martyn Andrews - Wooldridge, Chapter 7 is good preliminary reading • 5 lectures on Time Series methods by Dr Simon Peters, Wooldridge Chapters 10-12 good preliminary reading, other references mentioned in lectures

More Related