Cointegration and the ECM - LearnEconometrics.com

1 Cointegration and the ECM Two nonstationary time series are cointegrated if they tend to move together through time . For instance, we have established that the levels of the Fed Funds rate and the 3-year bond rate are nonstationary, whereas their differences are stationary. In the opaque terminology used in the time series literature, each series is said to be integrated of order 1 or I(1). If the two nonstationary series move together through time then we say they are cointegrated. Economic theory would suggest that they should be tied together via arbitrage, but that is no guarantee, so we perform a formal statistical test.

2 The test procedure is very simple. Regress one I(1) variable on another using least squares. Then test the residuals for nonstationarity using the (augmented) Dickey-Fuller test. If the series are cointegrated, the Dickey-Fuller test statistic will be statistically significant. The null hypothesis is that the residuals are nonstationary. Rejection of this leads to the conclusion that the residuals are stationary and the series are cointegrated. time series Plots qui tsline f, name(f, replace) qui tsline , name(df, replace) yline(0) qui tsline b, name(b, replace) qui tsline , name(db, replace) yline(0) graph combine f df b db, cols(2) This produces the set of graphs 051015federal funds rate1985q11990q11995q12000q12005q12010q1 date-2-101federal funds rate, D1985q11990q11995q12000q12005q12010q1dat e246810123-year Bond rate1985q11990q11995q12000q12005q12010q1 date-2-10123-year Bond rate, D1985q11990q11995q12000q12005q12010q1dat eNeither series looks stationary in levels.

3 Their general trend is downward. The differences do look stationary. I see no trend in the differences and both are centered about zero. Proceed with the formal stationarity tests. Here are the DF-GLS test results for the levels: The lags appear to be fairly short (actually, 1 in both cases). The statistic is in the 5% rejection region for both series (lag=1). That doesn t bode well for the rest of the example, which relies on these series to be random walks. A Phillips-Perron test, which also has a null-hypothesis of unit root, is performed for both series .

4 We include the trend option since the level of each appears to have a deterministic downward trend (though this certainly could not be true in the population, could it? I mean, can interest rates go down forever?). In neither case could the unit root null be rejected at 5% level. So, we will assume that b and f are I(1). Here are some details about the pperron command in Stata. Min MAIC = at lag 1 with RMSE .4793635 Min SC = at lag 1 with RMSE .4793635 Opt Lag (Ng-Perron seq t) = 3 with RMSE .4644163 1 2 3 4 5 6 7 8 9 10 11

5 12 [lags] Test Statistic Value Value Value DF-GLS tau 1% Critical 5% Critical 10% Critical Maxlag = 12 chosen by Schwert criterionDF-GLS for b Number of obs = 91 . dfgls bMin MAIC = at lag 1 with RMSE .3535111 Min SC = at lag 1 with RMSE .3535111 Opt Lag (Ng-Perron seq t) = 1 with RMSE.

6 3535111 1 2 3 4 5 6 7 8 9 10 11 12 [lags]

7 Test Statistic Value Value Value DF-GLS tau 1% Critical 5% Critical 10% Critical Maxlag = 12 chosen by Schwert criterionDF-GLS for f Number of obs = 91 and the results for each series Engle-Granger Test The test described below is commonly referred to as the Engle-Granger test. Regress b on f and a constant, save the residuals then use these in an augmented Dickey-Fuller regression. Manually, this is done regress b f predict ehat, residual regress , noconstant in dfuller uses additional lags of the first-difference variable.

8 To account for serial correlation, whereas the augmented Dickey-Fuller test implemented variable was generated by a stationary process. pperron uses Newey-West standard errors hypothesis is that the variable contains a unit root, and the alternative is that the pperron performs the Phillips-Perron test that a variable has a unit root. The nullDescription Statistics > time series > Tests > Phillips-Perron unit-root testMenu varname may contain time - series operators; see tsvarlist. You must tsset your data before using pperron; see [TS] tsset. lags(#) use # Newey-West lags regress display regression table trend include trend term in regression noconstant suppress constant term Main options description pperron varname [if] [in] [, options]SyntaxMacKinnon approximate p-value for Z(t)

9 = Z(t) Z(rho) Statistic Value Value Value Test 1% Critical 5% Critical 10% Critical Interpolated Dickey-Fuller Newey-West lags = 4 Phillips-Perron test for unit root Number of obs = 103.

10 Pperron f, trendMacKinnon approximate p-value for Z(t) = Z(t) Z(rho) Statistic Value Value Value Test 1% Critical 5% Critical 10% Critical Interpolated Dickey-Fuller Newey-West lags = 4 Phillips-Perron test for unit root Number of obs = 103.