13. Time Series Analysis: Asymptotics Weakly Dependent and Random Walk Process. Strict Exogeneity

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1 Outline: Further Issues in Using OLS with Time Series Data 13. Time Series Analysis: Asymptotics Weakly Dependent and Random Walk Process I. Stationary and Weakly Dependent Time Series III. Highly Persistent Time Series V. Correcting for Unit Root VI. Dynamically Complete Model Read Wooldridge (013, Chapter 11, 18. I. Stationary and Weakly Dependent Time Series Strict Exogeneity What have we learned? 1 Assume TS.1 TS.6, OLS has exactly same desirable properties that we derived for cross sectional data. If the errors are not drawn from a normal distribution, then we must rely on the central limit theorem assume large sample properties. 3 In time series analysis, observations tend to correlate across time TS.5. 4 Strict exogeneity (TS.3 is very restrictive. It might not be applicable to static and distributed lag models. Strict exogeneity (TS.3 may not make sense. Consider the static model rgdp t ms t + u Strict exogeneity (TS.3 assuming that u t is uncorrelated with regressors (X for all time periods, which implies that E(u t..ms t 1,,ms t, ms t+1, = 0 E(rgpd t.. ms t 1,,ms t, ms t+1, + 0 ms t TS.3 rules out the correlation between u t and ms t+1. Thus, the dependence of rgdp t on ms t+1 is not allowed. This may not make sense in time series analysis. I. Stationary and Weakly Dependent Time Series 3 I. Stationary and Weakly Dependent Time Series 4

2 Relax Exogeneity 4 Concepts Relax: assume TS.3 that x t and are weakly dependent. With large sample size, assume that u t is uncorrelated with regressors (x t in the same time period only. Goal: to have a general understanding of these concepts: 1. stationarime series. nonstationarime series 3. covariance stationary process 4. weakly dependent time series I. Stationary and Weakly Dependent Time Series 5 I. Stationary and Weakly Dependent Time Series 6 1. Stationary Time series. Nonstationary Time Series It refers to the stochastic process whose joint distributions are stable over time. Example x t = rgdp t {x t : t = 1,,...} The distribution x t has the same distribution as x 1. The joint distribution of (x 1, x must be the same as the joint distribution of (x 5, x 6 and must be the same as the joint distribution of (x t, x t+1. A stochastic process that is not stationary is said to be a nonstationary process Violation: a process with a time trend (trending series. It s not stationary. Its mean changes over time. Thus, stationary implies that the x t s are identically distributed and that the nature of any correlation between adjacent terms is the same across all periods. I. Stationary and Weakly Dependent Time Series 7 I. Stationary and Weakly Dependent Time Series 8

3 3. Covariance Stationary Process 4. Weakly Dependent Time Series It refers to a stochastic process has a finite second moment [E(x < ]. This is a weaker form of stationary. A stochastic process is covariance stationary if i E(x t is constant ii VAR(x t is constant iii Cov(x t, x t+h depends on h, not on t (for all t, h 0. Thus, this weaker form of stationarity requires onlhat the mean and variance are constant across time, and the covariance just depends on the distance across time A stationarime series process is weakly dependent if x t and x t+h are almost independent. as h increases without bound. Stationary versus Weakly Dependence Time Series Stationary joint distribution of a process Weakly dependence places restrictions on how strongly related x t and x t+h can be as h gets large. I. Stationary and Weakly Dependent Time Series 9 I. Stationary and Weakly Dependent Time Series 10 Weakly Dependent Time Series Example: Moving Average process of Order 1: MA(1 A covariance stationary process is weakly dependent if x t = e t e t 1 ; t=1,, Corr(x t, x t+h 0 as h. where {e t : t = 0, 1, } is i.i.d sequence with mean zero and variance e. [we write e t i.i.d (0, e ] In other words, covariance stationary sequences are asymptotically uncorrelated. mean: E(x t = 0 variance: Var(x t = (1+ 1 e covariance: Cov(x t, x t+1 = 1 e correlation: Corr(x t, x t+1 = 1 /(1 Question: Is x weakly dependent? I. Stationary and Weakly Dependent Time Series 11 I. Stationary and Weakly Dependent Time Series 1

4 Properties: MA(1 Example: Autoregressive Process of Order 1: AR(1 x t = e t e t 1 Properties: 1 x t is a weighted average of e t and e t 1 MA(1 process is weakly dependent! 3 Only adjacent terms are correlated. 4 After two or more time periods, x are independent! = 1 where {e t } is i.i.d (0, e and < 1 (stability condition Moments: E( =E( 1 when E( =0 VAR( = y = e /(1 Cov(, +h = h y Corr(, +h = h Question: Is y weakly dependent? I. Stationary and Weakly Dependent Time Series 13 I. Stationary and Weakly Dependent Time Series 14 Properties: AR(1 Example: Return from SET (Security Exchange of Thailand = 1 return t = return t 1 Properties: When { } is a stable AR(1 process (i.e., <1, it is asymptotically uncorrelated or weakly dependent. Let = 0.9 corr(, +1 = = 0.9 corr(, +0 = h = 0.1 where{e t } is i.i.d (0, e return t the rate of weekly return from SET index at time t. If < 1, the AR(1 process is weakly dependent. t = 0.067return t 1 (s.e. (0.038 [t stat] [1.75] [prob.] [.0795] n = 689, R = , R bar =.000 I. Stationary and Weakly Dependent Time Series 15 I. Stationary and Weakly Dependent Time Series 16

5 Trends Revisited A trending series cannot be stationary, since the mean is changing over time. T.S.1 : Linearity and weakly dependence The stochastic process {(x t1, x t,,x tk, : t = 1,, n} follows the linear model: A trending series can be weakly dependent. x t1 + + k x tk + u t If a series is weakly dependent and is stationary about its trend, we will call it a trend stationary process. As long as a trend (t is included, all is well. We add that {x t, } is stationary and weakly dependent. TS. : No Perfect Collinearity No independent variable is constant or a perfect linear combination of the others. I. Stationary and Weakly Dependent Time Series Assumptions: TS.1 TS.5 OLS Estimators are consistent. TS.3 : Zero Conditional Mean E(u t x t = 0 where x t ={x t1, x t,,x tk } are contemporaneously exogenous. Theorem 11.1: Consistency of OLS If TS.1 TS.3 hold, then OLS estimators are consistent. plim = j (j=0,1,,k Now we do not put restrictions on how u t is related to X in other periods. We relax the strong assumption on strict exogeneity of x tj. 19 0

6 Example : Consider the Static Model: inf t ms t + gdp t + u t Notes: 1 E(u t ms t, gdp t = 0 (weakly dependence u t are uncorrelated with ms t and gdp t (TS.3 TS.3 allows the correlation between u t 1 and ms t. That is, corr(u t 1, ms t 0 and corr(inf t 1, ms t 0. This kind of feedback is allowed under TS.3 :Change in ms t depends on last month s inflation (inf t 1. The implies that the TS.3 allow the feedback from inf this period on future values of ms. Example: Consider the AR(1 model 1 + u t Notes: 1 1 < 1is the stability condition for weakly dependence. Assume the errors have a zero expected value, given past values of y. E(u t 1,, = 0 E( 1,, = E( 1 1 Once y lagged one period has been controlled for, no further lags of y affect the expected value of 3 Strict exogeneity is violated. TS.3 requires that, for all t, u t is uncorrelated with each of (y 0, y 1,, y n 1. This can not be true. Show: Cov(,u t = Var(u t > 0 1 Assumptions: TS.1 TS.5 Asymptotic Normality TS.4 : Homoskedasticity VAR(u t x t = for all t The errors are contemporaneously homoskedastic, conditional on regressors only at time t. TS.5 : No Serial Correlation E(u t u s x t, x s = 0 for all t s We condition only on regressors only at time t and s. Theorem 11.: Asymptotic Normality of OLS If TS.1 TS.5 hold, OLS estimators are asymptotically normally distributed. In addition, OLS t, F and LM statistics are valid. 3 4

7 Example: Efficient Markets Hypothesis Example: Efficient Markets Hypothesis Let be weekly percentage return on NYSE composite index (from Wed close to Wed close = return t Specifhe AR(1 model return t return t 1 H 0 : 1 = 0 If H 0 is true, then the efficient markets hypothesis is valid. Efficient Markets Hypothesis E( 1,, = E( Intuition : past information are useless for prediction. If above relationship is false, we could use information on past returns to predict current return. (Assume the errors have a zero expected value, given past values of y. t = return t 1 (s.e. (0.081 (0.038 [t stat] [.5] [1.55] n = 689, R =.0035, R bar =.000 Interpretation: 1 Explain t = 1.55 (p value=.118. There is a weak (no evidence against the efficient markets (null hypothesis. 5 6 AR(1: Efficient Markets Hypothesis Dependent Variable: RETURN Method: Least Squares Sample(adjusted: Included observations: 689 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. C RETURN( R-squared Mean dependent var Adjusted R-squared S.D. dependent var.1154 S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic EM Hypothesis and AR( return t return t 1 + return t If the EM hypothesis is valid under H 0 : 1 =0 and =0 return t = return t return t (s.e. (0.081 (0.038 (0.038 {t} {.30} {1.58} {1.00} n = 688 R =.0048 R bar = Test overall significance: H 0 : 1 = 0, = 0 F statistic = 1.65; p value = Do we reject the efficient markets hypothesis? 7 8

8 AR(: Efficient Markets Hypothesis Dependent Variable: RETURN Method: Least Squares Sample(adjusted: Included observations: 688 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. C RETURN( RETURN( 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 Review: Weakly dependent time series { ; x t } implies that OLS inference procedures are valid. In the case AR(1, the condition is < 1. Terms: Highly persistent strong dependence time series. This series is not weakly dependent Highly persistent time series A random walk process Example: AR(1 model with = 1. A random walk process is = 1 : y at time t 1 : y at previous value e t : i.i.d. (0, e = 1 By repeated substitution, we can write as = e t e 1 + y 0 This is a highly persistent time series. This stochastic process { } is called a random walk. e t : i.i.d. (0, e This innovations e t for all t are independently identically distributed with mean zero and constant variance e. Assume y 0 is nonrandom; mean and variance are Mean: E( = E(y 0 Variance: Var ( = t e This shows that the process can not be stationary! 31 3

9 A random walk displays highly persistent behavior. A random walk is not covariance stationary. Reasons: Idea : the value oday is highly correlated with the value of y in the very distant future. 1 Variance is not constant. It depends on time t. hat is h periods from can be written as +h = +h 1 +e t+h +h = e t+h +h for all h 1 E(+h Var (+h = E( = (t+h e Cov(, +h = t e Corr(, +h = [t/(t+h] 1/ Mean: E(+h = E(. This means that it does not matter how far in the future we look, the best prediction is today s value y. Var(+h = (t+h e For an, h>1, covariance depends, not only on h but also on the starting point t. Cov(, +h = t e Corr(, +h = [t/(t+h] 1/ A random walk is not covariance stationary. Reasons: A unit root process. 3 Properties of corr(, +h t corr (y t, y t h t h For large t, corr 1 For fixed t as h, corr 0, but it does not do so quickly. For large t, the more slowlhe correlation tends to zero as h. For large h, we can choose a large enough t such that corr=1 Assuming that {e t } is a weakly dependent series. = 1 This is called a unit root process. Random walk is a special case of a unit root process 35 36

10 Random Walk vs. Weak Dependence Trending vs. highly persistent Example : GDP time series (Random Walk vs. Weak Dependence Weakly dependence: If GDP next year is weakly or not related to GDP thirty years ago. This series is not highly persistent or not strongly dependent. Highly persistence: If GDP next year is highly correlated to GDP from many years. This is strongly dependent. Key feature of { } that has a random walk or unit root process: The value of oday is highly correlated with y even in the distant future. A trending time series and highly persistent time series are different. Examples of Highly persistent time series: inflation rates, interest rates, and unemployment rates. Example of trending time series: GDP A series can be highly persistent and has a clear trend Random Walk with Drift Random Walk with Drift + 1 (* 0 is the drift term and e t is i.i.d. (0, e Mean: E( t implies that if 0 > 0 the expected value is growing overtime. if 0 < 0 the expected value is shrinking overtime. By repeated substitution, we write (* as +h is hat is h periods from (Random Walk with drift +h h +h +h t e 1 + y 0 Mean: E( t if y 0 =0 Variance Var( = t e E(+h h + E( h + Var (+h = (t+h e Cov(, +h = t e Corr(, +h = [t/(t+h] 1/ Interpretation: The best prediction of +h is plus the drift 0 h. Variance and Covariance remain the same

11 Unit Root Process with intercept I(0 versus I(1 Unit Root Process with intercept Weakly dependent processes are said to be integrated of order zero, I( Assume {e t } is any weakly dependent process. Here we obtain a whole class of highly persistent time series that also have linearlrending means. Unit root processes, such as a random walk are said to be integrated of order one, I(1 Note that a strong dependence or highly persistent time series violates weakly dependence assumption (TS.1 Thus usual inference procedures (t test, etc. are invalid. If we assume TS.1 TS.6 (CLM assumptions, strong dependence in regression analysis poses no problem, but this is unrealistic! 41 4 Informal Rules: Differencing or Not Estimate of Deciding whether a time series is I(1: Informal Rules Find the estimate of (rho When AR(1 process is stable, If < 1, then the process is I(0 If = 1, then the process is I(1 = Corr(, 1 For us, If > 0.8, differencing is warranted. This sample correlation is called the first order autocorrelation of {y}. = Corr(, 1 Eviews : View/Correlation to find called 43 44

12 Example: Wages and Productivity First Order Autocorrelation log(hrwage log(outphr t + t + u t log(hrwage log(outphr t + t + u t hrwage : average hourly wage outphr : output per hour t : time trend log( = log(outphr t.018t (s.e (0.37 (0.09 (.0017 {t stat} { 14.3} {17.65} { 10.4} n = 41 R = R bar= Interpretation: 1 Test and Interpret Test and Interpret 3 Interpret R Squared. = Corr(lhrwage t, lhrwage t 1 LHRWAGE LHRWAGE_1 LHRWAGE LHRWAGE_ = Corr(loutphr t, loutphr t 1 LOUTPHR LOUTPHR_1 LOUTPHR LOUTPHR_ Regress lhrwage on loutphr and t Dependent Variable: LHRWAGE Method: Least Squares Sample: 1 41 Included observations: 41 Variable Coefficient Std. Error t-statistic Prob. C LOUTPHR T 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 0 Example Unrealistic: = loutphr t t Find : 1 st order autocorrelation = for detrended log(hrwage = for detrended log(outphr (differencing is warranted Dependent Variable: LHRWAGE Method: Least Squares Date: 09/15/1 Time: 05:11 Sample: 1 41 Included observations: 41 Variable Coefficient Std. Error t-statistic Prob. C T 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 Dependent Variable: LOUTPHR Method: Least Squares Date: 09/10/11 Time: 18:33 Sample: 1 41 Included observations: 41 Variable Coefficient Std. Error t-statistic Prob. C T 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

13 Example Steps in finding log(outphr for detrended In the group window, View/Correlation = Corr(lhrwage_detrend t, lhrwage_detrend t-1 LHRWAGE_DETREND LHRWAGE_DETREND_1 LHRWAGE_DETREND LHRWAGE_DETREND_ Step 1: Regress log(outphr on time trend. Eviews: loutphr c t Step : Obtain residuals from regression in step 1. Choose Proc/Make Residuals and name resid01 as loutphr_detrend = Corr(loutphr_detrend t, loutphr_detrend t-1 Step 3: Generate a new series in Group. Eviews: loutphr_detrend_1 = loutphr_detrend( 1 LOUTPHR_DETREND LOUTPHR_DETREND_1 LOUTPHR_DETREND LOUTPHR_DETREND_ Step 4: Find sample correlation between loutphr_detrend and loutphr_detrend_1. In the group window, View/ Correlation Testing for unit root with no time trend 1. Testing for unit root with no time trend. Testing for unit root with linear time trend We know that the presence of a unit root implies that a shock toady has a long lasting impact. Terms: highly persistent time series strong dependent series I(1 series Informal Rule to decide whether a series is unit root or I(1, if the first order autocorrelation > 0.8, differencing is warranted 51 5

14 Testing with unit root with no time trend Unit Root: =1. = + -1 Consider the AR(1 model (* = + 1 where y 0 is the observed initial value. We assume that {e t } denote a process that has zero mean, given past observed y: It has a unit root if and only if =1. Hypothesis Testing H 0 : = 1; { } has a unit root or { } is I(1. H 1 : < 1; { } is I(0 or weakly dependent. E(e t 1,,,y 0 = 0 (18.18 Given (18.18, {e t } is said to be a martingale difference sequence with respect to { 1,, }. If {e t } is assumed to be i.i.d. sequence with zero mean and is independent of y 0, then it also satisfies ( Notes 1 In practice, 0< <1 <1. Usually no problem with a unit root. 3 >1 This case is not considered, as { } is explosive Let = 1 Dickey Fuller (DF Test A convenient equation for carrying out the unit root test is The asymptotic distribution of the t statistic under null hypothesis is calculated by Dickey Fuller (1979 known as Dickey Fuller distribution. The asymptotic distribution of the t statistic under H 0 has come to be known as the Dickey Fuller (DF test for a unit root. = + 1 Hypothesis Testing H 0 : = 0 { } is I(1 H 1 : < 0 { } is I(0 The problem is that, under H 0, 1 is I(1. Thus, the t statistic does not have approximate standard normal distribution even in large sample. Rejection rule: < c where c =.86 is the critical value at the 5 % level in Table 18.. If c= is used, we would reject H 0 more than 5% of the time when H 0 is true. Table 18. Asymptotic Critical Values for Unit Root t Test: No Time Trend level 1%.5% 5% 10% DF Test- No time trend c t-test C

15 Example: Wages and Productivity Test unit root no time trend log(hrwage log(outphr t + t + u t hrwage : average hourly wage outphr : output per hour t : time trend Tests for a unit root (no time trend: 1 lhrwage t = + lhrwage t 1 loutphr t = + loutphr t 1 Example: Test for a unit root: log(hrwage t = lhrwage t 1 {t stat} {4.619} { Notes 1. The coefficient on lhrwage shows that the estimate of is = 1 + = = Is it statistically less than one? What can you say about unit root? c = 3.43 ( =1% < c ( 4.19 < 3.43 We reject H 0 : =0 in favor of H 1 : < 0 at the 1% significance level. level 1%.5% 5% 10% DF Test- No time trend c What is p value for the coeffecient on lhrwage t 1? p value = (use Eviews Example: Easy way in Eviews Test lhrwage for a unit root. Choose View/Unit Root Test lhrwage t = + lhrwage t 1 Steps 1. Go to Workfile Open the variable of interest. (i.e., open lhrwage. Choose View/Unit Root Test. 3. In the Unit Root Test Box Test type: Augmented Dickey Fuller Test for unit root in: level Include in test equation: intercept lag length, User specified 0 Null Hypothesis: LHRWAGE has a unit root Exogenous: Constant Lag Length: 0 (Fixed t Statistic Prob.* Augmented Dickey Fuller test statistic Test critical values: 1% level % level % level *MacKinnon (1996 one sided p values. Augmented Dickey Fuller Test Equation Dependent Variable: D(LHRWAGE Included observations: 40 after adjusting endpoints Variable Coefficient Std. Error t Statistic Prob. LHRWAGE( C R squared Mean dependent var

16 Example: continue Test loutphr for a unit root. Choose View/Unit Root Test Null Hypothesis: LOUTPHR has a unit root Exogenous: Constant t = + loutphr t 1 t = loutphr t 1 {t stat} {3.83} { 3.40} {prob.} {0.0168} Lag Length: 0 (Fixed t-statistic Prob.* Augmented Dickey-Fuller test statistic Test critical values: 1% level % level % level Questions. 1 What is the estimate of? What can we say about unit root? *MacKinnon (1996 one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(LOUTPHR Included observations: 40 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. LOUTPHR( C R-squared Mean dependent var Test unit root with time trend (t Test unit root with t When { } has a clear trend = + t + 1 A trending stationary process has a linear trend but is I(0 about its trend. In running a DF regression, it can be mistaken for a unit root process. Thus, we should control for a time trend in DF. = + t + 1 H 0 : = 0 H 1 : < 0 Rejection rule: < c where c = 3.41 is the critical value at the 5 % significance level in Table H 0 : = 0, has a unit root (H 0 is rejected. Thus, = + t. This implies that the change in y has a mean linear in t. H 1 : < 0, { } is a trend stationary process. Table 18.3 Asymptotic Critical Values for Unit Root t Test: Linear Time Trend level 1%.5% 5% 10% DF Test- No time trend c DF Test with time trend c t-test C

17 Example: Wages and Productivity Test unit root with time trend ( t log(hrwage log(outphr t + t + u t hrwage : average hourly wage outphr : output per hour t : time trend Tests for a unit root (no time trend: 1 lhrwage t = + t + lhrwage t 1 loutphr t = + t + loutphr t 1 Example: augmented DW Test with linear time trend log t = t log(hrwage t 1 (s.e. (.0418 ( (.0334 {t} {1.8} { 1.91} {.470} [prob] [.08] [.0639] [.641] n=40 R = Analysis: 1 What is the estimate of? Is there any evidence of unit root at the 10% level? level 1%.5% 5% 10% DF Test with time trend c Interpret coefficient on time trend (Note that t statistic is invalid Easy way in Eviews lhrwage t = + t + hrwage t 1 Steps 1. Open series "lwagehr".. Choose View/Unit Root Test In the Unit Root Test Box Test type Augmented Dickey Fuller Test for unit root in Level Included in test equation: trend and intercept lag length, User specified 0 67 lhrwage t = + t + hrwage t-1 Null Hypothesis: LHRWAGE has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Fixed t Statistic Prob.* Augmented Dickey Fuller test statistic Test critical values: 1% level % level % level *MacKinnon (1996 one sided p values. Sample(adjusted: 41 Included observations: 40 after adjusting endpoints Variable Coefficient Std. Error t Statistic Prob. LHRWAGE( C t R squared Mean dependent var

18 t = + t + loutphr t-1 Null Hypothesis: LOUTPHR has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Fixed t-statistic Prob.* Augmented Dickey-Fuller test statistic Test critical values: 1% level % level % level *MacKinnon (1996 one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(LOUTPHR Included observations: 40 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. LOUTPHR( C t R-squared Mean dependent var Which model do you prefer? Model I log( = log(outphr.018t (s.e (0.37 (0.09 (.0017 {t stat} { 14.3} {17.65} { 10.4} n = 41 R = R bar= log(hrwage has a unit root! or I(1 log(outphr has a unit root! or I( V. Correcting for Unit Root Remedy: transformation I(0 Remedy: transformation I(0 Example: A a random walk process = 1 where {e t } i.i.d. (0,, Weakly dependent processes are said to be integrated of order zero I(0. = 1 = e t is an i.i.d sequence with with mean zero and constant variance. The first difference is I(0 Goal: Transform to get a unit root process that is weakly dependent by first difference. ( = 1 Example: Assume that e t is any weakly dependent process. This is is a unit root process. The first difference = e t is I(0 or weakly dependent. V. Correcting for Unit Root 71 V. Correcting for Unit Root 7

19 Remedy: transformation I(0 Consider log( that is I(1 log( = log( 1 where {e t } is any weakly dependent process. The first difference log(y is weakly dependent or I(0! log(y = log( log( 1 = e t Example Example: Efficient Markets hypothesis Model return t + return t 1 where return t = (price price( 1/price( 1 log(price Note that we use return t instead of price t in testing efficient markets hypothesis. Note that log(y [(y y( 1]/y( 1 V. Correcting for Unit Root 73 V. Correcting for Unit Root 74 Another benefit of differencing: It removes any linear time trend. Which model do you prefer? Consider a model t + v t E(v t = 0 1 (t 1 + v t 1 = 1 + v t E( = 1 E( v t = 0 E( = 1 is a constant! Differencing removes any linear time trend. Model I log( = log(outphr.018t (s.e (0.37 (0.09 (.0017 {t stat} { 14.3} {17.65} { 10.4} n = 41 R = R bar= Model II log( = log(outphr (s.e. (0.4 (0.173 [t stat] [.867] [4.67] n = 40; R = 0.364; R bar= V. Correcting for Unit Root 75 V. Correcting for Unit Root 76

20 Remedy: First differences to generate I(0 processes log( = log(outphr (s.e. (0.4 (0.173 [t stat] [.867] [4.67] n = 40; R = 0.364; R bar= Interpret: 1 Interpret the coefficient on log(outphr Interpret R squared. Notes: log(hrwage is the growth in average hourly wage. Eviews Notes: you can write log(hrwage as lhrwage lhrwage( 1 or D(lhrwage After differencing: regress lhrwage lwrwage( 1 on loutphr loutphr( 1 Dependent Variable: LHRWAGE-LHRWAGE(-1 Method: Least Squares Sample(adjusted: 41 Included observations: 40 after adjusting endpoints Coefficie Variable nt Std. Error t-statistic Prob. C LOUTPHR-LOUTPHR( 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 V. Correcting for Unit Root 77 V. Correcting for Unit Root 78 After differencing: Note that D(z = z z( 1 regress D(lhrwage on D(loutphr VI. Dynamically Complete Model Dependent Variable: D(LHRWAGE Method: Least Squares Sample(adjusted: 41 If the model is dynamically complete, there is no serial correlation. Included observations: 40 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. C D(LOUTPHR A dynamically complete model is a model that is correctly specified. R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Some think that all models should be dynamically complete. Serial correlation in the errors of a model is a sign of misspecification of the model. Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic V. Correcting for Unit Root 79 VI. Dynamically Complete Model 80

21 Simple Static Regression Model Simple Static Regression Model z t + u t z t + u t For consistency, we need E(u t z t = 0 To gain insight, we can write above equations as Generally, {u t } is serially correlated. But assume E(u t z t, 1, z t 1,,, = 0 E(u t z t, 1, z t 1,,, = 0 E( z t, 1, z t 1,,, = E( z t z t (1TS.5' holds. ( z t is contemporaneously exogenous. Interpretation: Once z t has been controlled for, no further lags of either y or z help to explain current y. Model 1 is dynamically complete. This implies that errors are serial uncorrelated. VI. Dynamically Complete Model 81 VI. Dynamically Complete Model 8 AR(1 model General model 1 + u t Assume that E(u t 1,,, = 0 E( 1,,, = E( 1 1 Note that this is the same thing as assuming that only one lag of y appears in E( 1,,, or there is no serial correlation. Interpretation: Once 1 has been controlled for, no further lags of y affect current y. Model is dynamically complete! x t1 + x t + + k x tk + u t x t = (x t1,.., x tk may contain lags of either z or y affecting y. Assume that E(u t x t, 1, x t 1,, = 0 E( x t, 1, x t 1,, = E( x t Interpretation: Further lags of either y or explanatory variables do not help to explain current y. Model 3 is dynamically complete! VI. Dynamically Complete Model 83 VI. Dynamically Complete Model 84

22 Recap on TS: Asymptotics Stationary and Weakly Dependent Time Series Asymptotic Properties of OLS Highly Persistent Time Series Formal Tests for Unit Root Dynamically Complete Model I. Stationary II. Properties III. Random Walk IV. Formal Tests V. Complete Model 85