Lectures on modelling non stationary time series

Transcription

1 Lectures on modelling non stationary time series by Roberto Golinelli 1. Univariate preliminary analysis 2 2. The stationarity issue in AR models: the unit root tests Unit roots and spurious regressions The dynamic specification (ARDL) Long run relationships and cointegrated variables Modelling systems Guidelines for the preparation of applied econometrics projects Reading list and acknowledgements 74 CIDE s PhD Lectures, Bertinoro (FO), June 2005 Department of Economics Strada Maggiore, Bologna (Italy) golinell@spbo.unibo.it UNIVARIATE PRELIMINARY ANALYSIS From a statistical p. o. v., a time series is a sequence of random variables ordered in time; we introduce the concept of STOCASTIC PROCESS (SP): {X t }, t = 1, 2,..., T The probability structure of a stochastic process is determined by its joint distribution. Example of a SP: the white noise model [1] x t = c + ε t ε t ~ n.i.d. (0, σ 2 ) x t is normally and independently distributed over time with constant variance and c mean (also constant). Q: IS IT AN APPROPRIATE MODEL FOR THE MACROECONOMIC TIME SERIES? Eviews/phil/series u (unemployment rate)/descript. stats Series: U Sample Observations 40 Mean Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability a corresponding artificial series can be generated with same sample mean and standard deviation of the historical u: genr uaswn = * nrnd genr meanline = 6.94 plot u uaswn meanline 2

2 U UASWN MEANLINE LQR LQRASWN MEANLINE The white noise (WN) model for the unemployment rate in Italy would state that: u randomly fluctuates around a constant mean (6.94) with constant variance ( ). But the white noise model does not fit actual data for u because it does not feature the time series most common characteristic: PERSISTENCE. In fact, the actual u is by far more persistent than the simple WN process under and above the natural rate of about 7%. Q: IS THIS RESULT PECULIAR TO UNEMPLOYMENT? Eviews/lqr/series lqr (logs of capacity utilisation ratio). plot lqr lqraswn meanline From the plot below it is evident that the capacity utilisation has a completely different path with respect to the unemployment rate: in fact, lqr is markedly less persistent than u. However, the capacity utilisation ratio still persists more than the corresponding artificial series (generated as a white noise realisation). A: WE HAVE TO FIND OTHER REFERENCE MODELS. MORE REALISTIC STATISTICAL MODELS ARE COMBINATIONS OF DIFFERENT ε; THEY ARE CALLED ARMA MODELS. Example of another SP: the AR(1) model [2] x t = c + α x t-1 + ε t ε t ~ n.i.d. (0, σ 2 ) is a WN The variable x t is not independently distributed over time because it depends on x t-1. In a model for u, we can estimate c and α parameters of equation [2] by using the OLS method: ls u c u(-1) Method: Least Squares Sample(adjusted): Included observations: 39 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. C U(-1)

3 Residual Actual Fitted from previous regression output we note that: the estimate of α parameter is very close to one; the AR(1) model fits unemployment quite well. Since residuals are estimates of ε t (white noise processes), we have to check the classical assumptions by using the diagnostic (mispecification) tests: Under the null: AR(1) residuals no autocorrelation rejected no heteroschedasticity not rejected normality not rejected We can react to autocorrelation by increasing to 2 the order of the AR process: the AR(2) model is written as [3] x t = c + α 1 x t-1 + α 2 x t-2 + ε t ε t ~ WN where there is one more parameter, and the dynamics is extended to the second lag. ls u c u(-1) u(-2) (results not reported) the residual tests are all fine (white noise errors); the AR(2) model equally fits well; the sum of the two α estimates is close to one. Now, also try with the lqr variable: ls lqr c lqr(-1) Dependent Variable: LQR Sample(adjusted): Included observations: 46 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. C LQR(-1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Durbin-Watson stat In this case, a first order model is enough to avoid residuals problems and the alpha estimate is equal to about 0.6 (< 1). Note that the dynamics of the capacity utilisation rate (less persistent than the unemployment rate) is more difficult to be fitted by the AR model (R 2 = 0.367, against 0.976). Preliminary findings: a) data persistence is explained by AR models; b) the sum of the AR parameter estimates is often close to one; c) the more persistent the path, the easier to fit the data by AR models and the closer to one is the sum of alpha estimates. In addition, note that not all the economic series are untrended, and in case of trended variables we must introduce deterministic components in our statistical models in order to (potentially) give account of this further feature. 6

4 7 8 From an economic p. o. w., the nature of previous u and lqr variables excludes the presence of a deterministic trend, being both measured by ratios. On the other side, there are many variables whose levels can continuously grow over time (output, real wages, prices, etc.). For example if we define logs of the real wage as: genr lwp = log(w/p), and if plot lwp, we can note that its path over time is trended, explained by some causal effects (e.g. labour productivity growth). The same apply to ly (logs of real output), equally trended, or consumer price levels p. Previous statistical models can be easily extended to this feature by including a deterministic trend (t); e.g. the equation [1] becomes: [1'] x t = c + β t + ε t and we can fit this WN plus deterministic trend model to actual lwp data: ls lwp The WN plus trend model does not fit data, and the regression residuals are very much persistent (strong positive autocorrelation). we have to introduce wage dynamics with the AR(1) plus trend model (an extension of the equation [2]): [2 ] x t = c + β t + α x t-1 + ε t ls lwp lwp(-1) Dependent Variable: LWP Method: Least Squares Sample(adjusted): Included observations: 39 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. C LWP(-1) The inclusion is very important indeed: thanks to dynamics the residuals are now fine (results not reported); the relevance of time trend vanishes while the autoregressive parameter estimate is close to one (as in many cases of AR model estimates). Some first tentative conclusions confirm previous preliminary findings: a) despite the inclusion of a deterministic trend, lwp persistence needs an AR dynamics in general, many economic series can be represented by AR models of different orders, with or without deterministic trends; b) the (sum of) AR parameter estimates is very often close to one Residual Actual Fitted

5 Q: WHAT DOES POINT B) IMPLY IN TERMS OF THE AR MODELS STATISTICAL PROPERTIES? The next step will be the study of the statistical properties of the AR models with (or without) unit roots: a unit root is found in the SP of an AR model when the sum of the alpha parameters is equal to one (necessary condition) THE STATIONARITY ISSUE IN AR MODELS: THE UNIT ROOT TESTS Consider the AR(1) model in the equation [2], and for the moment let s ignore the deterministic components: [2 ]x t = α x t-1 + ε t ε t ~ n.i.d. (0, σ 2 ) is a WN By introducing the lag operator: Lx t = x t-1 L 2 x t = LLx t = Lx t-1 = x t-2 L 0 x t = x t we can redefine equation [2]: x t = α Lx t + ε t (1 - αl) x t = ε t x t = ε t /(1 - αl) and if α < 1 we have that: 1/(1 - αl) = 1 + αl + α 2 L 2 + α 3 L = α i L i i=0 a geometric series converges if the absolute ratio of successive terms is less than [2 * ]x t = (1 + αl + α 2 L 2 + α 3 L ) ε t = ε t + αε t-1 + α 2 ε t-2 + α 3 ε t In equation [2*] the AR(1) process is written in the corresponding MA( ) form (Wold representation). E(x t ) = 0 (this result depends on the absence of deterministic components); Var(x t ) = E[x t -E(x t )] 2 = E[ε t + αε t-1 + α 2 ε t-2 + α 3 ε t ] 2 = E[ε 2 t + α 2 ε 2 t-1 + α 4 ε 2 t-2 + α 6 ε 2 t ] = σ 2 [1 + α 2 + α 4 + α ] = σ 2 /(1 - α 2 ) Var(x t-k ) Cov(x t,x t-k ) = E{[x t -E(x t )] [x t-k -E(x t-k )]} = E{[ε t + αε t-1 + α 2 ε t α k ε t-k + α k+1 ε t-k-1 + α k+2 ε t-k ] [ε t-k + αε t-k-1 + α 2 ε t-k ]} = α k σ 2 /(1 - α 2 ) = α k Var(x t ) α k Var(x t-k )

6 Autocorrelation coefficient of order k ρ k = Cov(x t,x t-k )/Var(x t ) = Cov(x t,x t-k )/Var(x t-k ) = α k If α < 1, the AR(1) model is STATIONARY since its moments do not depend on t. The autocorrelation coefficient ρ k decreases when k increases (the memory of the process decreases with k). Example: if α = 0.6 (as in the case of the AR(1) model for the capacity utilisation ratio in logs) then: x t = ε t ε t ε t ε t ε t ε t ε t after six periods, the shock is no longer economically significant. An easy way to appreciate the path of the shock is to draw the impulse-responses function of a series. IMPULSE-RESPONSES IN THE STATIONARY AR(1) MODEL responses horizon timing impulse shocked x i = x' i = x' i - x i 0 t s x' t =x t + s s 1 t t+2 0 x' t+1 =αx' t +ε t+1 = αx t +αs+ε t+1 = x t+1 +αs x' t+2 =αx' t+1 +ε t+2 = αx t+1 +ααs+ε t+2 = x t+2 +α 2 s αs α 2 s 3 t α 3 s h t+h 0... α h s 11 The responses decrease because α < 1. Example: the AR(1) model for the capacity utilisation ratio. Eviews/lqr/quick/estimate VAR/lqr 1 1/impulse h=10 (multiple graphs) Response of LQR to One S.D. LQR Innovation What is depicted is the path of a transitory shock: given that lqr variable is explained by a stationary AR(1) model, 0.6 <1, the response to the impulse vanishes over time. A transitory shock can be interpreted as a demand shock: an increase in demand (positive shock) causes a short run increase in output, but leaves unaffected the long run potential output of the economy (given by the supply side). Contrast the stationarity ( α < 1) case with the unit roots case (α = 1 in equation [2 ]): x t = x t-1 + ε t Repeated backwards substitution allows to write: 12

7 x t = x 0 + ε t + ε t-1 + ε t-2 + ε t ε 2 + ε 1 where x 0 is assumed to be a fixed initial value for the process. In a process with unit roots, second moments depend on time (non stationarity): E(x t ) = x 0 t Var(x t ) = E[x t - x 0 ] 2 = E[ ε i ] 2 = t σ 2 i= 1 Cov(x t,x t-k ) = E{[x t - x 0 ] [x t-k - x 0 ]} = E{[ε t + ε t-1 + ε t ε t-k + ε t-k-1 + ε t-k ε 2 + ε 1 ] [ε t-k + ε t-k-1 + ε t-k ε 2 + ε 1 ]} = = (t-k) σ 2 Var(x t-k ) ρ k = Cov(x t,x t-k )/[Var(x t ) Var(x t-k )] 0.5 = (t-k)σ 2 /σ 2 [t(t-k)] 0.5 = t k = t lim Var(x t ) = ; lim Cov(x t,x t-k ) = ; lim ρ k = 1 ; t t t Eviews/phil/quick/estimate VAR/u 1 2/impulse h=20 (multiple graphs): the unemployment rate in Italy Response of U to One S.D. U Innovation While in the stationary case a shock (innovation) ε has an effect on x that diminishes with t (transitory shock), in the unit root case ε has a sustained (permanent) effect. In the case of the unemployment rate in Italy, a 0.5% shock is very persistent: after 20 years it is still 0.5% (permanent shock). The unit root model has an infinite memory. DETERMINISTIC COMPONENTS Previous outcomes do not substantially change by adding deterministic components to the AR model: y t = d t + x t where: d t are the deterministic variables, and x t is a zero mean AR(1) process. By using the Wold representation of the AR(1) model: y t = d t + ε t /(1 - αl) (1 - αl) y t = (1 - αl) d t + ε t y t = α y t-1 + d t - α d t-1 + ε t case (a): d t = µ 0 (only the constant term) y t = (1-α) µ 0 + α y t-1 + ε t by defining (1-α) µ 0 = c, we have the equation [2]. case (b): d t = µ 0 + µ 1 t (linear trend) y t = α y t-1 + µ 0 + µ 1 t - α [µ 0 + µ 1 (t-1)] + ε t = = αy t-1 + µ 0 + µ 1 t - αµ 0 - αµ 1 t + αµ 1 + ε t = = (1-α)µ 0 +αµ 1 + µ 1 (1-α)t + αy t-1 +ε t by defining (1-α)µ 0 +αµ 1 = c, and µ 1 (1-α) = β we have the equation [2 ]. Summary of two useful models (a) y t = c + α y t-1 + ε t (b) y t = c + β t + α y t-1 +ε t 14

8 (a) E(y t ) = µ 0 E(y t ) = E(d t ) + E(x t ) = d t (b) E(yt ) = µ 0 + µ 1 t Var(y t ) = E[y t -E(y t )] 2 = E(x t ) 2 = Var(x t ) Cov(y t,y t-k ) = E{[y t - E(y t )] [y t-k - E(y t-k )]} = E{x t x t-k } = = Cov(x t,x t-k ) The deterministic variables in y t only change the mean (that in any case is non stochastic); second moments are the same as those of x t (zero mean) variable. the stationary condition is still α < 1 if α < 1: model (a) is a stationary AR(1) model with mean µ 0 0 (MEAN REVERTING) model (b) is a stationary AR(1) plus trend model (TREND REVERTING) if α = 1: model (a): y t = y t-1 + ε t is the RANDOM WALK model (b): y t = µ 1 + y t-1 + ε t is the RANDOM WALK WITH DRIFT for: stationary non drifting time series drifting trend stationary (TS) time series for: non drifting difference stationary (DS) time series drifting difference stationary (DS) time series When α = 1 we talk about random walks because the non stationary AR is a first order model. 15 We can summarise the univariate unit root concept by following Johansen (1997), who notes that the specific unitroot model: x t = x t-1 + ε t or: x t = ε t can be also written as: x t = x o + ε 1 + ε ε t This is the random walk: a person that starts walking from a square (x o ), and takes steps (ε i ) of random size and direction: x t is his position after t steps when he starts at x o. By modelling a variable by a random walk, we do not try to reproduce its sample path, because we decide that these details are not important to explain (in fact, we model them as random) only the qualitative behaviour of the path matters: it is a float (once it has reached a level, it stays there until it reaches a new level). On the other hand, in the model: unpredictable (random) part x t = π x t-1 + ε t or: x t = α x t-1 + ε t predictable part of the movement π (= α - 1) represents the glue of the process; if π 0 (then, α 1) neighbouring values of x t are more often close together, and we got a wave-like behaviour. While if: π -1 (then, α 0), neighbouring values of x t are almost unrelated (independent). In general, when -2 < π < 0, or -1 < α < 1, the path of x t exhibits a mean reversion. 16

9 Summary exercise: Practice with the stationary AR(1) model Use the model (a) above: y t = c + α y t-1 + ε t with: ε t ~ n.i.d. (0, σ 2 ) under stationary condition α < 1 we have that E(y t ) = E(y t-1 ) = µ 0 We can simpler summarise first and second moments. E(y t ) = c + α E(y t-1 ) + E(ε t ) hence: µ 0 = c /(1 - α) If we substitute this definition in model (a), and take the expected value of the square: (y t - µ 0 ) = α (y t-1 - µ 0 ) + ε t E(y t - µ 0 ) 2 = α 2 E(y t-1 - µ 0 ) 2 + E(ε t ) 2 Var(y t ) = γ 0 = σ 2 /(1 - α 2 ) Finally, multiply the demeaned equation times (y t-k - µ 0 ): (y t - µ 0 ) (y t-k - µ 0 ) = α (y t-1 - µ 0 ) (y t-k - µ 0 ) + ε t (y t-k - µ 0 ) if we define: E[(y t - µ 0 ) (y t-k - µ 0 )] = γ k then: γ k = α γ k and: ρ k = α ρ k-1 = α k ρ 0 = α k (note that ρ 0 = 1) Simulation analysis can be used to verify a number of stylised facts (procedure: simular1.prg). In order to simulate an AR(1) model, we need to set three parameters: µ 0, α, and the ratio = µ in order to o obtain the three genuine parameters of the AR(1) model: c = µ 0 (1 - α) α σ = ratio µ 0 (1 - α 2 ) ½ In the procedure µ 0 =%0 α=%1 ratio=%2 sign of α=%3 (0=positive, 1=negative) γ 0 17 ' %0 = mean (id number) ' %1 = alpha (e.g or > 60) ' %2 = ratio between s.d.(y)/mean(y) (e.g. 50% ---> 50) ' %3 = sign of alpha (0=positive, 1=negative) scalar s = %2/100*%0*(1-( (-1)^%3 *%1/100)^2)^.5 smpl rndseed %0 genr e%0%1%2_%3 = s*nrnd ' set the initial value (random number from long run mean & variance of y ' alternatively you can start from zero or deterministically from the mean %0 smpl genr y%0%1%2_%3=%0 + e%0%1%2_%3/(1-( (-1)^%3 *%1/100)^2)^.5 ' iterate the other values smpl genr y%0%1%2_%3 = %0*(1-((-1)^%3*%1/100)) + (-1)^%3*%1/100*y%0%1%2_%3(-1)+ e%0%1%2_%3 smpl With alternative simulations we can assess... (a) the issue of the initial value of the dynamic processes what happens if we start far away from the mean? (e.g ) (b) that the persistence of the path of y t depends on the α parameter (with 0 α < 1 ): it grows with α 1 (c) what happens when -1 α < 0 (e.g /40/80/ /1) (d) what role played by ratio Hints: open a new working file your_name.wf1 (quarterly data from to ) open the program simular1.prg run various scenarios (in parentheses above) compare plots, correlograms, impulse-response functions save all useful results 18

10 HOW TO TEST DS VS TS MODELS The data generating process (DGP) is DS if α = 1, while it is TS (or mean reverting) if α < 1. The main inferential point is to find a significance-test for α estimates. The testing models are two: model (a) includes a constant term, model (b) includes both a constant and a trend. Models (a) and (b) can be conveniently reparametrised in order to ease inference: (a) y t = c + π y t-1 + ε t (b) y t = c + β t + π y t-1 +ε t where: π = α - 1 THE DICKEY-FULLER (DF) UNIT ROOTS TEST H 0 : π = 0 α = 1 H 1 : π < 0 α < 1 y t variable is a random walk y t variable is a stationary AR(1) Under the null, y t is first order integrated: I(1) because it is stationary after one difference. Under the alternative it is a mean reverting AR(1) in model (a) or a trend stationary variable in model (b); in both cases y t is I(0) because it is stationary without (zero) difference. The choice of the deterministic components [i.e. model (a) or (b)?] depends on the economic nature of the variable: model (a) for ratios and rates, model (b) for levels; the historical pattern: model (a) for non drifting variables (remember that level variables are often drifting). 19 Sometimes, residuals from models (a) or (b) are autocorrelated: if so, they tell that first-order autoregressive dynamics is not enough. In these cases we have to pass from the DF (first-order) test to the Augmented Dickey-Fuller, the ADF(p) test analyses a more general p+1 order dynamics. The corresponding (a) and (b) models for ADF(p) testing are: p (a) y t = c + π y t-1 + γ i y t-i + ε t i= 1 (b) y t = c + β t + π y t-1 + p i= 1 γ i y t-i + ε t Note that the augmentation is made in order to obtain white noise residuals; the augmentation is mainly suggested for high frequency observations (e.g. monthly, quarterly). The rule of thumb states the augmentation of the DF test is often quite similar (or equal) to data periodicity. Example. If the variable y t is quarterly, it is appropriate to start with a fourth-fifth order dynamics. Suppose we are using the model (a), and that a fourth order dynamics is appropriate to explain the path of the variable under scrutiny [AR(4) model with constant and without trend]: y t = c + α 1 y t-1 + α 2 y t-2 + α 3 y t-3 + α 4 y t-4 + ε t it can be conveniently rearranged as follows: y t -y t-1 = c + α 1 y t-1 - y t-1 ± α 2 y t-1 ± α 3 y t-1 ± α 4 y t-1 + α 2 y t-2 ± α 3 y t-2 ± α 4 y t-2 + α 3 y t-3 ± α 4 y t-3 + α 4 y t-4 + ε t 20

11 21 22 y t = c + (α 1 +α 2 +α 3 +α 4-1) y t-1 - (α 2 +α 3 +α 4 ) y t-1 - (α 3 +α 4 ) y t-2 - α 4 y t-3 + ε t this specification coincides with ADF(3) testing model by defining: π = (α 1 +α 2 +α 3 +α 4-1); γ 1 = - (α 2 +α 3 +α 4 ); γ 2 = - (α 3 +α 4 ); and γ 3 = - α 4 ; H 0 : π = 0 α 1 +α 2 +α 3 +α 4 = 1 y t is I(1); no a random walk, but simply DS H 1 : π < 0 α 1 +α 2 +α 3 +α 4 < 1 y t is I(0) stationary AR(4) model The critical values of the ADF(p) test are the same as the DF test. The choice of the starting augmentation order depends on: data periodicity (see above) significance of γ i estimates white noise residuals After preliminary estimation, non significant parameter augmentation can be dropped in order to enjoy more efficient estimates. For this reason, Campbell-Perron (1991) intuitively suggest a dropping down procedure from p max. Then, such procedure has been supported (and refined) by the findings of Hall (1994), and Ng-Perron (1995, 2001). In particular, simulations carried out e.g. in Ng-Perron (1995) show a strong association between the choice of p and the severity of size distortions (over-rejections) and/or the extent of power loss (too few rejections). Some ADF unit-root test applications The unemployment data for Italy Eviews/phil/u/view/line graph/unit root test/levels/intercept ADF Test Statistic % Critical Value* % Critical Value % Critical Value *MacKinnon critical values for rejection of hypothesis of a unit root. Dependent Variable: D(U) Method: Least Squares Sample(adjusted): Included observations: 38 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. U(-1) D(U(-1)) C R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) The same test is accomplished for u first differences: ADF Test Statistic % Critical Value* % Critical Value % Critical Value *MacKinnon critical values for rejection of hypothesis of a unit root. Augmented Dickey-Fuller Test Equation Dependent Variable: D(U,2) Method: Least Squares Sample(adjusted): Included observations: 37 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. D(U(-1)) D(U(-1),2) C R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

12 The unemployment rate in Italy is generated by a statistical process with one unit root: u is I(1), and u is I(0). This fact is apparently impossible, since u is a ratio limited between zero and 100%; the same can be said with reference to other ratios or rates (interest rates, the inflation rate, etc.). An explanation comes by quoting, among the others, Hall, Anderson, Granger (1992, note 5): «The conclusion that yields to maturity are integrated processes can not be true in a very strict sense because integrated series are unbounded, while nominal yields are bounded below by zero. Nevertheless it is evident from the data that the statistical characteristics of yields are closer to those of I(1) series than I(0) series, so that for the purposes of building models of the term structure it is appropriate to treat these yield series as if they were I(1)». Practice: does previous result change if we take u in logs (variable lu) instead of in levels? Try both reference models (a) and (b); do the answers change with models? Another application: the logs of capacity utilisation in Italy. ADF Test Statistic % Critical Value* % Critical Value % Critical Value *MacKinnon critical values for rejection of hypothesis of a unit root. Augmented Dickey-Fuller Test Equation Dependent Variable: D(LQR) Method: Least Squares Sample(adjusted): Included observations: 45 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. LQR(-1) D(LQR(-1)) C R-squared Mean dependent var -4.54E-05 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) The logs of capacity utilisation ratio are I(0), as also suggested by the profile of the impulse-responses. Example: Do the Treasury bills interest rates have a unit root? Eviews/termine/ plot rbot3 rbot6 rbot ADF(12) test for rbot3 (levels): RBOT3 RBOT6 RBOT12 ADF Test Statistic % Critical Value* % Critical Value % Critical Value *MacKinnon critical values for rejection of hypothesis of a unit root. ADF(12) test for d(rbot3) (first differences): ADF Test Statistic % Critical Value* % Critical Value % Critical Value *MacKinnon critical values for rejection of hypothesis of a unit root. The variable rbot3 is I(1), for an explanation see above and read the following phrase. 24

13 «Yet, interest rates are almost certainly stationary in levels. Interest rates were about 6% in ancient Babylon; they are about 6% now. The chances of a process with a random walk component displaying this behaviour are infinitesimal. Pr( r 1991 <100% r 4000BC = 6%) it is infinitesimal if r are or contain a random walk; it is near one if interest rates are an AR(1) with a coefficient of 0.99», Cochrane (1991, p. 208). Last sentence introduce the issue of the relevance of the time span, rather than the number of observations. Given an economic variable, e.g. the inflation rate, different data periodicity imply different time spans (inflation data, see below): name periodicity time span # of observ. lypc lpq 4 70q1-97q4 112 (1) (2) lpm 12 72m1-98m7 319 Given the number of observations T, the wider the time span, the higher the power of unit roots tests (case 1). Given the time span, high frequency data do not relevantly improve the power of the tests (case 2). The power gain from increasing the data span is bigger than the power gain from increasing the sample size while leaving the data span fixed. It would be surprising if simple time disaggregation helped in the estimation of long run relations. Hendry (1986, OBES). An example: Is the inflation rate I(1) or I(0)? If we use annual data (lypc.wf1), we have: LPC and in fact, the ADF test results are: DLPC ADF Test Statistic % Critical Value* % Critical Value % Critical Value *MacKinnon critical values for rejection of hypothesis of a unit root. Augmented Dickey-Fuller Test Equation Dependent Variable: D(LPC) Sample(adjusted): Included observations: 104 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. LPC(-1) D(LPC(-1)) D(LPC(-2)) D(LPC(-3)) C While, in first differences, the null is 1% rejected: ADF Test Statistic % Critical Value* % Critical Value % Critical Value *MacKinnon critical values for rejection of hypothesis of a unit root. Result: the inflation rate is I(0), and price levels are I(1). But a completely different picture emerges with high frequency data (quarterly, lpq.wf1, and monthly, lpm.wf1); in fact, the common outcome is that inflation rate is I(1), and price levels I(2). Understandable, if we look at the plots below: 26

14 *DLP D4LP D12LP 12*DLP THE ELLIOTT-ROTHEMBERG-STOCK (DF-GLS) UNIT ROOTS TEST Fact: While the presence-absence of a unit root has important implications, many remain skeptical about the conclusions drawn from such tests why? remember Size of the test: probability that the test actually rejects the null when the null is true. Power of the test: probability that the test correctly rejects the null when the alternative is true. The ADF test suffers from severe size distortions (overrejection of the unit root hypothesis) when the movingaverage polynomial of the first differenced series has a large negative root. The ADF test has low power when the root of the autoregressive polynomial is close to but less than unity Elliott, Rothemberg, Stock (1996) find that local GLS detrending of the data yields substantial power gains. Ng, Perron (2001) show that size and power may be further improved when the truncation lag is appropriately selected (e.g. with their specific MAIC p-selection rule). 27 DF-GLS and ADF share the same alternative univariate models: (a) without or (b) with a deterministic linear trend; and test for the same hypotheses: H 0 : y t has a unit root H 1 : y t is stationary The DF-GLS test is accomplished in two steps. 1 st step: GLS detrending. case (a) case (b) y (α) 1 = y 1 ; c (α) 1 = 1 ; y (α) 1 = y 1 ; c (α) 1 = 1 ; t (α) 1 = 1 y (α) t = y t -α * y t-1 for t = 2, 3,.., T y (α) t = y t -α * y t-1 for t = 2, 3,.., T c (α) t = 1 - α * for t = 2, 3,.., T c (α) t = 1 - α * for t = 2, 3,.., T t (α) t = t - α * (t-1) for t = 2, 3,.., T where α * = 1-7/T where α * = /T OLS estimates of the δ 0 and δ 1 parameters y (α) t = δ 0 c (α) t + e t y (α) t = δ 0 c (α) t + δ 1 t (α) t + e t y d t = y t - δ ^0 detrended y t is defined as y d t y d t = y t - ( δ ^ + δ ^ t) 2 nd step: ADF tests of the detrended series y d t. p y d t = π y d t-1 + γ i y d t-i + ε t i= 1 The DF-GLS test corresponds to the Student-t of the OLS estimate of π in the equation above (note that all the deterministic variables are excluded, and the equation is the same in both cases). While DF-GLS t-ratio follows the ADF distribution (but without constant) in the case (a), the asymptotic distribution differs when case (b) is considered (c.v. are simulated in the ERS paper)

15 Concluding remarks (Stock and Watson, ch. 12, 2002) In case the dependent variable and/or regressors are non stationary, then autoregressive coefficients are biased towards zero, OLS t-statistics nonnormal under the null, spurious regression. Hence: the conventional hypothesis tests, confidence intervals, and forecasts are unreliable. The precise created by the nonstationarity, and the solution to that problems depend on its nature. Main sources of nonstationarity are trends and breaks. Trend: a persistent long run movement of a variable over time. It is of two types: deteministic (nonrandom function of time) stochastic (random, varies over time) (examples: ly and dlpc in lypc.wf1) Many econometricians think it is more appropriate to model economic time series as having stochastic rather than deterministic trends. Economics is a complicated stuff. It is hard to reconcile the predictability implied by a deterministic trend with the complications and surprises faced year after year by workers, businesses, and governments (... examples...). For these reasons, our treatment of trends in economic time series focuses on stochastic rather than deterministic trends. (p. 458) Break: arises when the population regression function changes over time over the course of the sample. In economics it occurs for changes in economic policy and/or in the structure of the economy, for inventions, etc. Usually, it entails changes in the regression parameters and poorer-than-expected forecasting performance. Again, the nature of the break suggests the best solution (switching/evolving parameter regressions) UNIT ROOTS AND SPURIOUS REGRESSIONS Most econometric analyses are based on sample variance and covariance estimates among variables. Non stationarity causes problems (unconditional moments are not defined): a likely result is spurious regression, and the use of standard large samples theory for valid estimation and inference in the linear model is not allowed. Historical background (since 1920s). Spurious correlation is an observed sample correlation between two series which, though appearing statistically significant, is a reflection of a common trend rather than a reflection of any genuine underlying association. Allen (1949, p. 156): There is a stronger positive correlation between the birth rate and the number of storks in Sweden, since each has been declining for various reasons. Correlation is a statistical concept which is neutral as regards causal relations (economic concept). The non stationarity cancels a number of standard statistical properties and tools. Consider the model: y t = c + β z t + ε t where we suppose that y t and z t are independent. Classical assumptions of regression: I. the regressors are either deterministic or stationary random variables (uncorrelated with the error term) II. E(ε t ) = 0; E(ε t ) 2 = σ 2 ; E(ε t ε t-k ) = 0 k > 0. 30

16 If both assumptions hold, then: H 0 : β = 0 Pr( t > 1.96) = 0.05 while, if y t and z t are I(1) variables, assumption I. clearly fails, and the effect on t-statistic distribution is: H 0 : β = 0 Pr( t > 1.96) (i.e. appearance of a false significant regression well over the 5% significance level: problems in the size of the test). In addition, such regressions are characterised by: 2 2 t (yt ŷt ) R = 1 is very high, close to one 2 t (yt y ) (since the variables are both trended, the ratio above can be very small) Very low Durbin-Watson (DW) statistic of 1 st order autocorrelation, close to zero. DW 2 (1 - ρ), where ρ is the 1 st order autocorrelation coefficient of the regression residuals. If DW 0 then ρ 1 and it suggests regression residuals are probably I(1). Example: Spurious regressions with artificial data. Eviews/new/workfile/quarterly/ (T=124). open/program/montecarlo.prg/ run it various times and check: t, R 2 and DW. 31 Suggested remedies in literature Granger-Newbold (1974, JE) suggest a rule of thumb to detect spurious regressions: when R 2 >> DW. the remedy they suggest is to impose a = (1-L) filter to I(1) series in order to make them stationary, and improve inference: ls D(Y) C D(Z) In this way, t-statistics are no longer significant (as expected, since the two variables were independently simulated), R 2 is close to zero, and DW test suggests non autocorrelated residuals (close to 2). Sims-Stock-Watson (1990, E) note that, in levels static regressions, t-statistics test the following hypothesis: H 0 : β = 0 y t = c + ε t is FALSE because y t is I(1) H 1 : β 0 is FALSE too, since y t and z t are not related both null and alternative hypotheses are false; this raises further inference problems. The main problem with these spurious regressions is that nothing in the model gives account of y t persistence (only the residuals, since z t is not related to y t ). the remedy they suggest is to add lags in order to reach white noise residuals (since the persistency is caught by the dynamic specification): ls Y C Z Y(-1) Z(-1) In addition, the true model (y is a random walk) is nested in the dynamic model. Things improve, but we still miss statistical foundations for inference (with I(1) variables, t- statistics are non standard and R 2 are uninformative). Some preliminary findings: dynamics matters very much (white noise residuals); always remember what model is underlying both the null and the alternative hypotheses. 32

17 Example of a crazy regression: the US consumers look at logs of UK incomes when they purchase goods (logs of US consumption)! Static model. Eviews/ardlusuk/ ls lcus c lyuk Dependent Variable: LCUS Sample(adjusted): 1959:1 1998:1 Included observations: 157 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. C LYUK R-squared Mean dependent var S.E. of regression Akaike info criterion Durbin-Watson stat Prob(F-statistic) If we impose first difference transformation: ls d(lcus) c d(lyuk) as suggested by Granger- Newbold (results not reported), we obtain positive and negative findings: non significant t-statistic and very low R 2 (positive side); we loose levels and the information of economic theory (negative side); residual autocorrelation (negative side). Reconsider now spurious regression for US consumption in the context of the dynamic model by augmenting the static regression with lags up to one year (four lags). ls lcus c lyuk(0 to 4) lcus(-1 to 4) as suggested by Sims-Stock-Watson (results not reported). Main findings: the residuals are white noise; the sum of lagged consumption parameter estimates is close to one, and the sum of UK income parameter estimates is close to zero THE DYNAMIC SPECIFICATION (ARDL) The problems related to non stationarity can be partly solved in a dynamic framework because many potentially right models are nested in it. Empirical analysis of level (long run) relationships has been an integral part of time series econometrics and pre-dates the literature on unit roots and cointegration, see Hendry- Pagan-Sargan (1984). The fundamental contribution of this literature is on the specification and estimation of level relationships, rather than testing for their presence, since co-integration theory was missing (not yet fully established). Q: HOW TO USE ECONOMIC THEORIES WHEN CONSTRUCTING AN EMPIRICAL MODEL? Two extreme approaches (see Granger, 1999, ch. 1): (1) Theory contains the only pure truth, so has to be at the basis of the model, leaving little place for stochastics, uncertainty or exogenous shocks to the system. (2) Theory is useless; better atheoretical models based just on examination of the data and using any apparent regularities and relationships found in it. Most applied economists take a middle ground, using theory to provide the initial specification (variables of interest) and then data exploration techniques to extend or refine the starting model, leading to a form that better represents data. Theory static statements about economic relations (long run relations) Reality (data) dynamic fashion 34

18 A: TO PRODUCE A BRIDGE FROM THE PRISTINE THEORY TO THE MORE PRAGMATIC DATA ANALYSIS When economic theory proposes an equilibrium relationship between two variables this may be seen as the long-run steady-state solution of a dynamic model. Define the simple 1 st order Auto-Regressive Distributed- Lags (ARDL) model: [4] y t = c + α 1 y t-1 + β 0 z t + β 1 z t-1 + ε t where ε t ~ n.i.d. (0, σ 2 ). A long run relation is something that the dynamic process would satisfy if all errors were switched-off, and the equations would then bring back the process to a set of values where the long run relation is satisfied (steady state). The long-run-steady-state-non-stochastic solution of eq. [4] is obtained by setting: y t = y t-1 = y * ; z t = z t-1 = z * ; ε t = 0 y * = c/(1-α 1 ) + (β 0 +β 1 )/(1-α 1 ) z * and the long run (level) relationship is measured by the parameter β = (β 0 +β 1 )/(1-α 1 ). Model [4] can be reparametrised in a convenient way in order to better understand the mechanism of adjustment towards the long run relationship: [4 ] y t = c+β 0 z t +(α 1-1) [y t-1 -(β 0 +β 1 )/(1-α 1 )z t-1 ] + ε t by defining α = (α 1-1) and, as above, β = (β 0 +β 1 )/(1-α 1 ), we obtain the specification of the error correction mechanism (ECM) model: [4 ] y t = c + β 0 z t + α [y t-1 - β z t-1 ] + ε t 35 given the level z t-1, y * t-1 = β z t-1 measures the target level, and [y t-1 - y * t-1] is the error(equilibrium)-correction term. The ECM form of the model may be seen as comprising the short-run transitory effect and the long run relationship, and describes how the long run solution is achieved via error correction feedback. In fact, if -2 < α < 0 (that corresponds to -1 < α 1 < 1) the equation [4 ] equilibrates in presence of a discrepancy between y t-1 and y * t-1: it guarantees that, in the long run, y will converge to its target y *. If y t is not on its long run path, and suppose that y t-1 > y * t-1 {y t-1 < y * t-1} from the ECM representation -2 < α < 0 ensures that there is pressure, from the error-correction term, for y t < 0 { y t > 0}. In other terms, if -2 < α < 0, in disequilibrium y t will move towards its long run path; both from above and below, and the movement will be in proportion of the last period s error given by [y t-1 - y * t-1]. α also measures the speed (and the path) of the adjustment of y t to disequilibrium. α > 0 the model is explosive α = 0 the model does not adjust -1 < α < 0 stable process of adjustment α = -1 the model adjusts in one period -2 < α < -1 overshooting adjustment α = -2 the model continuously oscillates α < -2 the model is explosive Big forecast problems in presence of level-breaking relationships in y * t (see Clements-Hendry, 1999). 36

19 Testing for the existence of a level-relationship It is more convenient to use another parametrisation of [4]: [4 * ] y t = c + β 0 z t + π 1 y t-1 + π 2 z t-1 + ε t where: π 1 = (α 1-1) = α; and π 2 = (β 0 +β 1 ) = -α β There are two testing possibilities: i. H 0 : π 1 = 0 (t-statistic); ii. H 0 : π 1 = π 2 = 0 (F-statistic). Note that t and F distributions are not standard; asymptotic critical values are tabulated by Pesaran-Shin-Smith (2001), and they depend on: the variables are either I(0) or I(1); the presence of the deterministic trend and/or the constant; the number k of explanatory (forcing) variables. Example: Unrestricted constant and no trend (5% asymptotic c.v.) t F k I(0) I(1) I(0) I(1) Main features of the ARDL approach: Developed in the field of the Auto Regressive Distributed Lags models (dynamic specification, Sargan, LSE, etc.). 37 The null hypothesis of both (t and F) tests is the absence of a long run relationship. Critical values are ad hoc tabulated: one set of c.v. are obtained by assuming that all the variables are I(0), the other set by assuming all the variables are I(1). In the case where z t and ε t are correlated, the ARDL procedure requires estimation of an augmented version of the original model. Hence, the important issue in the application of the ARDL procedure is the choice of the order of the distributed lag function on y t and z t. Example: Does exist a long run stable relationship between consumption and income in the US? Eviews/ardlusuk/ardl genr dlcus = d(lcus) genr dlyus = d(lyus) ls dlcus c dlcus(-1 to 3) dlyus(0 to -3) lcus(-1) lyus(-1) Both t and F tests do not reject H 0 spurious relation? Probably it s better to think in terms of omitted variables e.g. the quarterly inflation rate: pius (Deaton consumption model) is added to previous ARDL specification: genr dpius = d(pius) ls dlcus c dlcus(-1 to 3) dlyus(0 to -3) dpius(0 to 3)lcus(-1) lyus(-1) pius(-1) Given LM(4) autocorrelated residuals, we added dlcus lags 4 and 5: the results are in /equation/ardlfinal/ ls dlcus c dlcus(-1 to 5) dlyus(0 to -3) dpius(0 to 3)lcus(-1) lyus(-1) pius(-1) (model used in estimating/testing for the long run relationship). 38

20 Both t and F tests reject H 0 either by using I(0) or I(1) c.v., since statistics are: t = -3.5 and F = 8.46 while corresponding I(0)-5% c.v. are t = -2.86, F = 3.79; and I(1)- 5% c.v. are t = -3.53, F = 4.85 (see Pesaran, Shin, Smith, 2001, p. T.2 and T.4). The estimates of the long run parameters are: βˆ y = = 0.94 ; βˆpi = = The estimate of the loading parameter is αˆ = : each quarter, lcus adjusts by about 10% towards the target (equilibrium) level given by: lcus * = 0.94 lyus 2.24 pius. Often the absence of a long run relationship is the symptom of the omission of relevant variables. In fact, the partial long run relationship (lcus-lyus) is very persistent (i.e. reverts slowly), while the additional information from pius path can further explain savings inertia, and form a cointegrated relationship together with (lcus-lyus). plot lcus-lyus pius pius lcus-lyus LONG RUN RELATIONSHIPS AND COINTEGRATED VARIABLES Ex ante, by pretesting with ADF test, suppose we know that y t and z t are I(1): does a long run relationship between y and z exist? are y and z co-integrated? Cointegration definition: Two integrated I(d) variables are co-integrated if there exists a linear combination of them which is integrated I(c), with c < d. The case d = 1 and c = 0 is interesting in that cointegration implies an ECM representation (EqCM is the Hendry s &co update) which allows to rewrite a dynamic model in levels I(1) as a dynamic model which involves only variables I(0). THE ENGLE AND GRANGER PROCEDURE: [A] OLS estimation of the static (cointegration) regression y t = c + β z t + u t Note that if the combination u t =y t (c+βz t ) is I(0), then the integrated y and z variables are also cointegrated. [B] Unit roots test of the cointegration regression residuals û t = y t - ( ĉ + βˆ z t ) [C] If y and z are cointegrated, then βˆ (OLS β estimator) is superconsistent. [D] Dynamic (short run) ECM modelling of y t, z t and û t. Note that under the hypothesis of cointegration all the variables in the ECM model are I(0). 40

21 RATIONALE OF STEPS [A, B]: COINTEGRATION AND COMMON TRENDS At univariate level, suppose: y t = µ 1t + v 1t where: µ 1t = µ 1 + µ 1t-1 + ε 1t ε 1t ~ i.i.d. (0, σ 2 11) v 1t = v 1 + α 1 v 1t-1 + ε 1t ε 1t ~ i.i.d. (0, σ 2 12) µ 1t is a random walk (the non stationary component of y t ), and v 1t is an AR(1) with α 1 <1 (the stationary component of y t ). The same is supposed for z t : z t = µ 2t + v 2t where: µ 2t = µ 2 + µ 2t-1 + ε 2t ε 2t ~ i.i.d. (0, σ 2 21) v 2t = v 2 +α 2 v 2t-1 + ε 2t ε 2t ~ i.i.d. (0, σ 2 22) In the static regression: y t = c + β z t + u t, we have that: u t = y t - β z t c = µ 1t + v 1t - β (µ 2t + v 2t ) c = = [µ 1t - β µ 2t ] + [v 1t - β v 2t ] c where: [µ 1t - β µ 2t ] are the I(1) components, and [v 1t - β v 2t ] are the I(0) components. The cointegration condition is [µ 1t - β µ 2t ] = 0 (when combined, the I(1) components of y and z variables cancel each other). Under the cointegration condition, the error term is u t = (v 1t - β v 2t c) ~ I(0) while, if the cointegration condition is not satisfied, u t ~ I(1). In other terms, u t ~ I(0) means that µ 1t = β µ 2t : the I(1) component of y t is the same as that of z t up to a scalar β, the parameter that converts µ 2t in µ 1t. µ 2t is the stochastic common trend of y t and z t ; in fact, under the cointegration assumption, we have that: z t = µ 2t + v 2t y t = β µ 2t + v 1t 41 µ 2t is the common source of nonstationarity; and, by substituting µ 2t definition in the cointegrated y t we have: y t = β (z t - v 2t )+ v 1t = c + β z t + (v 1t β v 2t c) Again, the static regression residuals are stationary (though autocorrelated) if y t and z t are cointegrated. INTUITION BEHIND STEP [C]: SUPERCONSISTENCY If y t and z t are cointegrated, the OLS method yields a superconsistent estimator of the cointegrating parameters since the effect of the common trend dominates the effect of the stationary component. The omission of the dynamics is not very relevant if the variables are I(1) and cointegrated. The cointegrated combination is a strong linear relationship, but with relevant biases in small samples (see Banerjee et al. (1993) results). Example with simulated data. Eviews/new/workfile/ undated/1 200/open/program/supercon.prg/: run the program, and it will display different dispersions around the regression lines in I(0), yi0 against zi0, and I(1), yi1 against zi1, variables. All data were simulated with long run parameter equal to 1, with both I(0) and I(1) variables, and adjustment parameter equal to ½. The regression output from I(1) variables ls yi1 c zi1 shows that 200 observation are enough to enjoy superconsistence of the OLS estimator of cointegrated relations. The recursive estimation gives the intuition of the issue: with samples < 100 observations the amount of the 42

22 bias is very relevant (this result confirms Banerjee et al. (1993) outcomes) YI ZI0 YI ZI1 43 Previous results with simulated I(1) time series suggests a further question: why the static (cointegration) regression does estimate the contegration (long run) parameter even though the DGP is dynamic?. Hypotheses: (i) y and z are I(1) and cointegrated; (ii) the true DGP is (see eq. 4 p. 33): y t =c+ α 1 y t-1 + β 0 z t + β 1 z t-1 + ε t [ε t ~nid(0,σ 2 )] Fact: if y and z are cointegrated, the OLS regression of y on z alone (with no lags) yields a slope βˆ that is a (super)consistent estimator of the long run parameter β = (β 0 +β 1 )/(1-α 1 ) because: the OLS criterion of picking βˆ to minimise the sum of squared residuals forces it towards [(β 0 +β 1 )/(1-α 1 )]. In fact, if β β 0 [(β 0 +β 1 )/(1-α 1 )] y t = β z t + u 1t The residuals of the static regression are autocorrelated (LM test and correlogram) but stationary. Following the Engle and Granger (1987) approach, white noise residuals are not essential at this stage. Save residuals and perform the ADF test without deterministic components (CRDW and CRDF with different c.v., see Engle-Granger, 1987). The same regression with I(0) variables ls yi0 c zi0 shows short term (and not long term) parameter estimate, and autocorrelated residuals; by adding the lagged dependent variable, we are able to find a consistent estimate of the long run parameter. while, if where u 1t = β 1 z t-1 +α 1 y t-1 +ε t β [(β 0 +β 1 )/(1-α 1 )] y t = β z t + u 2t ; where u 2t = β 1 /(α 1-1) z t + α 1 /(α 1-1) y t + ε t /(1-α 1 ) Given the hypotheses (i) and (ii), u 1t ~ I(1) and u 2t ~ I(0). Since the sum of squares of an I(1) variable increases without size as the sample goes to infinity, the OLS estimator will pick the β estimate such that the corresponding residuals are closer to an estimate of u 2t instead of u 1t. Of course, this fact per se does not prevent u 2t from being autocorrelated.