Lecture 8a: Spurious Regression

1 Lecture 8a: Spurious Regression1 Old Stuff The traditional statistical theory holds when we run regressionusing (weakly or covariance ) stationary variables. For example, when we regress one stationary series onto anotherstationary series, the coefficient will be close to zero andinsignificant if the two series are independent. That is NOT the case when the two series are two independentrandom walks, which are nonstationary2 Spurious Regression The Regression is Spurious when we regress one random walk ontoanother independent random walk. It is Spurious because theregression will most likely indicate a non-existing coefficient estimate will not converge toward zero (thetrue value). Instead, in the limit the coefficient estimate willfollow a non-degenerate t value most often is typically very # by construction y and x are two independent random walksy = rep(0,n)x = rep(0,n)ey = rnorm(n)ex = rnorm(n)rhoy = 1rhox = 1for (i in 2:n) {y[i] = rhoy*y[i-1] + ey[i]x[i] = rhox*x[i-1] + ex[i]}4 Result of Spurious Regressionlm(formula = y ~ x)Coefficients:Estimate Std.

2 Error t value Pr(>|t|)(Intercept) < 2e-16 ** **The estimated coefficient ofxis , and is significant, despitethat the true coefficient is zero. This result mistakenly indicates thatyandxare related; actually they are independent of each Flags for Spurious residual is highly persistent. In theory, the residual of aspurious has unit result changes dramatically after the lagged dependentvariable is added as new regressor. The previously significantcoefficient will become : always check the stationarity of theresidual. The Regression is Spurious if the residualis nonstationary (cannot reject the null hypothesisof the unit root test)7 What causes the Spurious Regression ?Loosely speaking, because a nonstationary series contains stochastic a random walkyt=yt 1+etwe can show its MArepresentation isyt=et+et 1+et 2+ stochastic trendet+et 1+et 2+:::causes the series toappear trending (locally).

3 Regression happens when there are similar local Series Plot of Simulated Data050100150200 12 10 8 6 4 20obsy9 The solid line isyand dotted line isx. Sometimes their local trendsare similar, giving rise to the Spurious : just because two series move together doesnot mean they are related!11 Lesson: use extra caution when you run regressionusing nonstationary variables; be aware of thepossibility of Spurious Regression ! Check whetherthe residual is 8b: Cointegration13 Definitionyandxare cointegrated if both of following are nonstationary;xis exists a linear combination ofyandxthat is stationaryIn short, two series are cointegrated if they are nonstationary TheoryMany economic theories imply cointegration. Two examples +b income+ interest rate=inflation rate+real interest rateExercise: can you think of another example?15 OLS Estimator is SuperconsistentUnlike the Spurious Regression , whenyandxare cointegrated,regressingyontoxmakes super sense.

4 Super because thecoefficient estimate will converge to the true value (which is nonzero)super fast (at rate ofT 1instead ofT 1=2).16 Simulation# y and x are nonstationary but related (cointegrated)y = rep(0,n)x = rep(0,n)ey = rnorm(n)ex = rnorm(n)beta = 4rhox = 1for (i in 2:n) {x[i] = rhox*x[i-1] + ex[i]y[i] = beta*x[i] + ey[i]}17 Result of Cointegration Regression > = lm(y~x)> summary( )Coefficients:Estimate Std. Error t value Pr(>|t|)(Intercept) <2e-16 **The coefficient ofxis , very close to the true value 4. Thisresult makes super sense (however, the t value does not followstandard t or normal distribution! More on this issue later).18 Residual is stationary for cointegrated series> = $res> ( , k = 1)Augmented Dickey-Fuller Testdata: = , Lag order = 1, p-value = hypothesis: stationaryThe p-value is less than , so rejects the null hypothesis of unitroot (rigorously speaking, the p-value is wrong here because it isbased on the Dickey-Fuller distribution.)

5 More on this issue later).19 Engle-Granger Test for CointegrationThe Engle-Granger cointegration test (1987, Econometrica) isessentially the unit root test applied to the residual of series are cointegrated if the residual has no unit series are not cointegrated (and the Regression is Spurious ) ifthe residual has unit rootThe null hypothesis is that the series are NOT of Engle-Granger Test You may think that Engle-Granger test follows the Dickey-Fullerdistribution. But that is true only when the coefficient isknowna priori. When is unknown and estimated by sample, the distributionchanges, as shown by Phillips and Ouliaris (1990, Econometrica). For example, when there is no trend in the cointegrationregression, the 5% critical value of the Engle-Granger test , rather than (the critical value for the Dickey-Fullerunit root test).21 SimulationStep 1: we need to generateyandxunder the null hypothesis,which is that they are not cointegratedStep 2: then we regressyontox;and save the residualStep 3: we regress the differenced residual onto its first lag.

6 Thet value of the first lagged residual is the Engle-Granger testStep 4: we repeat Steps 1,2,3 many times. The 5% percentile ofthe distribution of the t values is the 5% critical value for theEngle-Granger test22 Lesson: make sure you are using the correct criticalvalues when running the cointegration Technical IssueConsider the cointegration regressionyt= 0+ 1xt+et(1)The superconsistency meansT( 1 1))Nonstandard DistributionSo we cannot conduct the hypothesis testing using the the consumption-income relationship, we may test the nullhypothesis that the marginal propensity to consume is unity, ie.,H0: 1= 1We can construct the t test as usual. However, we cannot compare itto the critical values of t distribution or normal OLS (DOLS) EstimatorStock and Watson (1993, Econometrica) suggest adding the leads andlags of xtas new regressorsyt= 0+ 1xt+p i= pci xt i+et(2)Now 1is called dynamic OLS estimator, and it is asymptoticallynormally : use DOLS estimator so that the normaldistribution can be used in hypothesis testing27 Lecture 8c: Vector Error Correction Model28 Big Picture of Multivariate can apply VAR if series are can apply VAR to differenced series if they are nonstationaryand not need to apply vector error correction model if series arenonstationary and cointegrated29 ExampleConsider a bivariate seriesytandxt.

7 Suppose they are cointegrated,then the first order vector error correction model is yt= 01+ y et 1+ 11 yt 1+ 12 xt 1+ut(3) xt= 02+ x et 1+ 21 yt 1+ 22 xt 1+vt(4) et 1=yt 1 0 0xt 1(5)where the residual etof the cointegration Regression is called errorcorrection term. It measures the deviation from the long that the residual etis stationary when the series arecointegrated. So it makes sense to use it to explain the stationary ytand will be omitted variable bias if we apply VAR todifferenced series when they are cointegrated; the omittedvariable is et is called error correction model because it shows how variableadjusts to the deviation from the equilibrium (the errorcorrection term) speed of error correction is captured by yand x:31 Engle-Granger Two Step ProcedureStep 1: Obtain the error correction term, or the residual of thecointegration Regression (5).Step 2: Estimate (3) and (4) using the second step, we can treat the estimated 0and 1as the truevalues since they are the variables in the error correction models are stationary ifthe series are cointegrated.

8 So the t test and f test follow thestandard particular, we can apply the standard test for GrangerCausality:H0: 12= 0H0: 21= 033 VECM-based Cointegration vector error correction model (VECM) can also be used toconstruct a test for null hypothesis of no cointegration isH0: y= 0; x= test follows nonstandard distribution because under the nullhypothesis the series are not cointegrated and et is possible that some of variables are error-correcting; othersare that case, the variables that are not error correcting are calledweakly exogenous. Exogenous because they are example, ifxis weakly exogenous, the VECM becomes yt= 01+ y et 1+ 11 yt 1+ 12 xt 1+ut xt= 02+ 21 yt 1+ 22 xt 1+vt35 Lecture 8d: Empirical Example36 Data We are interested in the relationship between the federal fundrate (sr), 3-month treasury bill rate (mr) and 10-year treasurybond rate (lr) The monthly data from Jan 1982 to Jan 2014 are downloadedfrom Fred Interest RatesTimeInterest Rates19851990199520002005201020152468101 21438We see overall the three series tend to move together (sr and mr areparticularly close).

9 Nevertheless we are not sure the co-movement (orsimilar local trends) indicates a Spurious Regression or a of of sr40lr and sr seem nonstationary as their ACF decays very Root Testob = length(lr) = c(NA, lr[1:ob-1]) # first lagdlr = lr - # differencesummary(lm(dlr~ ))Coefficients:Estimate Std. Error t value Pr(>|t|)(Intercept) . **The adf test with zero lag is , greater than the 5% critical So the null hypothesis of unit root cannot be rejected for Regression > = lm(lr~sr)> summary( )Coefficients:Estimate Std. Error t value Pr(>|t|)(Intercept) <2e-16 ** <2e-16 **So 0= 2:51249; 1= 0:80281:Those are super-consistent Correction TermThe estimated error correction term is the residual, and is computedas et=lrt 0 1srtThe R command isect = $resExercise: how to interpret a positive error correction term? Howabout an extremely negative error correction term?

10 44 Exercise: how to obtain the DOLS estimator of 0and 1?45 Engle-Granger Test for lr and sr> ob = length(ect)> = c(NA, ect[1:ob-1])> dect = ect - > summary(lm(dect~ ))Coefficients:Estimate Std. Error t value Pr(>|t|)(Intercept) **The Engle-Granger test is , greater than the 5% critical So the null hypothesis of no cointegration cannot be result is not surprising. Basically it says that the relationbetween the federal fund rate and 10 year rate is very weak (not asstrong as cointegration). Given no cointegration, the appropriatemultivariate model is VAR in in difference for lr and srlm(formula = dsr ~ + )Estimate Std. Error t value Pr(>|t|)(Intercept) ** **lm(formula = dlr ~ + )Estimate Std. Error t value Pr(>|t|)(Intercept) ** seems that sr does not Granger cause lr. Does this result makesense?