Introduction to Time Series Data
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1 Topic 8 Introduction to Time Series Data ARE/ECN 240 A Graduate Econometrics Professor: Òscar Jordà
2 Outline of this topic Dependence and covariance-stationarity, ergodicity and CLT for MDS Wold representation theorem Autoregressive models Properties Forecasting Estimation Moving average models
3 Dependent Samples So far we have assumed availability of random i samples, i.e. fy i ;X i g n i:i:d: i=1 with y i ;X i» D(¹; Ð) Now, the joint density for fy 1 ;:::;y T g is: f(y 1 ; :::; Y T jx; ) 6=f(y 1 jx; ):::f(y T jx; ) =f(y 1 jx; )f(y 2 jy 1 ;X; )::: f(y t jy t 1 ;:::;y 1 ;X; ):::f(y T jy T 1 ; :::; y 1 ;X; ) Key: find a way to limit the amount of dependence on the past, limit the memory
4 Example: Okun s Law Recall, Okun s law relates changes in GDP with changes in the unemployment rate. But GDP today is correlated with past GDP. Suppose that y t = x t + u t i:i:d u t = ½u t 1 + ² t ; ² t» D(0;¾ 2 ) Hence y t 1 = x t 1 + u t 1 so the residual process is (y t x t ) = ½(y t 1 1 x t 1 ) + ² t or in regression form: y t = x t + ½y t 1 x t 1 ½ + ² t y t = x t + ½y t 1 + x t 1 ± + ² t
5 Okun s law continued Therefore, if serial correlation is limited to one period, f(y t jy t 1 ;:::;y 1 ;X; ) =f(y t jy t 1 ;X; ) and dhence f(y 1 ;:::;y T jx; ) =f(y 1 jx; ) T t=2f(y t jy t 1 ;X; ) which except for the marginal density for the first observation, looks like the usual maximum likelihood.
6 STATA Example: Okun s Law. reg dy du Source SS df MS Number of obs = 243 F( 1, 241) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = dy Coef. Std. Err. t P> t [95% Conf. Interval] du _cons estat dwatson Durbin-Watson d-statistic( 2, 243) =
7 Residual Serial Correlation. corrgram e LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor]
8 STATA Example (cont.). reg dy du l1.dy l1.du Source SS df MS Number of obs = 242 F( 3, 238) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = dy Coef. Std. Err. t P> t [95% Conf. Interval] du dy L du L _cons estat dwatson Durbin-Watson d-statistic( 4, 242) =
9 Residual Serial Correlation. corrgram e LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor]
10 Covariance stationarity A basic concept in time series meant to give limits to the amount of memory that is acceptable. fy t gis covariance (weakly) stationary if E(y t )=¹ is independent of t and cov(y t ; y t j ) = j is independent of t for all j. (j) is called the autocovariance function. Notice that corr(y j t ; y t j ) = = ½ j 0 ½(j) is called the autocorrelation function
11 Ergodicity fy t gis strictly stationary if the joint distribution of (y t ; :::; y t j ) is the same as that for (y s ; :::; y s j ) for any t and j. Ergodicity: a stationary time series is ergodic if (j)! 0asj!1 Ergodic Theorem: (LLN) If fy t g is strictly stationary and ergodic with Ejy t j < 1 then as T!1 1 X T p yt y t! E(yt t ) T t=1
12 Ergodic Theorem (cont.) If in addition E(y t 2 ) < 1 then as T!1 T ^¹ p! ¹ ^¹ = 1 TX y t ; ¹ = E(y t ) p T ^ t=1 j! j j p ^½ j! ½j ^ j = 1 TX (y t ^¹)(y t j ^¹) T j j t=j+1
13 Common Transformations gdp u q1 1960q1 1970q1 1980q1 1990q1 2000q1 2010q1 qtr 1950q1 1960q1 1970q1 1980q1 1990q1 2000q1 2010q1 qtr -5 0 d dy du q1 1960q1 1970q1 1980q1 1990q1 2000q1 2010q1 qtr q1 1960q1 1970q1 1980q1 1990q1 2000q1 2010q1 qtr
14 Common Transformations Detrending: Linear trend: y t = + t + y t First difference transformation Growth rate transformation: log(y t )=log(y t ) log(y t 1 ) s-difference transformation: s-growth transformation y t = y t y t 1 s y t = y t y t s s log(y t ) = log(y t ) log(y t s )
15 Seasonal Adjustments Dummy variable adjustments: SX y t = s d s t + y t with d s t =1ift 2 season s; 0otherwise s=1 X11; X12: these are seasonal filters used by statistical ti ti agencies when they report seasonally adjusted data.
16 Wold representation theorem Any covariance-stationary series y t can be written as an infinite moving-average process MA( 1) of its innovation process or prediction error, i.e. 1 1X y t = ¹ + à j ² t j ; with the convention à 0 =1 and j=1 E(² t )=0;E(² t ² t j )=¾ 2 < 1 if j =0; 0otherwise Sometimes also called the impulse response representation. Notice: IR(y t+j ; ² t =1) = E(y t+j j² t =1; F t 1 1) ) E(y t+j j² t =0; F t 1 1) ) = à j
17 Autoregressions The simplest time series model, the AR(1) y t = + ½y t 1 + ² t! E(y t jf t 1 )= + ½y t 1 if E(² t jf t 1 )=0 Definition: ² t is a martingale difference sequence if E(² t jf t 1 ) =0 Recall, previously correct specification meant Now E(² i jx i )=0! E(Xi² 0 i )=0 E(² t jf t 1 )=0! E(y t j ² t )=0foranyj>0 0
18 AR(1) zero-mean Model Suppose f² t gis i.i.d. with E(² t )=0;E(² 2 t )=¾ 2 < 1 y t = ½y t 1 + ² t = ½(½y t 2 + ² t 1 )+² t we will need the = ½y condition j to t 2 + ² t + ½² j½j < 1 t 1 limit the memory. = ::: If j½j = 1then we have X t 1 = ½y infinite memory. 1 + ½ j ² t j j=0 If j½j > 1 then the 1X process is explosive. = ½ j ² t j Wold representation j=0
19 Properties of the AR(1) If j½j < 1 then fy t g is strictly stationary and 1 ergodic and 1X E(yt t )= ½ j E(² t )=0 V (y t )= Lag operator notation: j=0 1X ½ 2j V (² t )= j=0 0 L j y t = y t j ¾2 1 ½ 2 y t = ½y t 1 + ² t =) (1 ½L)y t = ² t =) y t = ² t 1 ½L = 1X 1X ½ j L j ² = j t ½ ² t j j=0 j=0 ² t
20 The AR(1) Impulse Response Function 1.2 Response of Y to Nonfactorized One Unit Y Innovation RHO = Response of Y to Nonfactorized One Unit Y Innovation RHO =
21 Forecasting with the AR(1) Recall y t = + ½y t 1 + ² t By recursive substitution: y T +h = + ½ + ::: + ½ h 1 + ½ h y T + ² T +h + ½² +h 1 + ::: + h 1 T ½ ² T For simplicity, assume =0, then ^y T (h) = E(y T +h jy T ) = ½ h y T Notice h 1 ^u T (h) = y T +h ^y T (h) = ² T +h + ::: + ½ h 1 ² T
22 Forecasting (cont.) Hence, the forecast error variance for the conditional mean forecast ^y T (h) = E(y T +h jy T ) is V (^u T (h)) = V (² T +h + ::: + ½ h 1 ² T ) = ¾ 2 (1+½ 2 + ::: + ½ 2(h 1) )
23 Statistical ti ti Properties of the AR(1) Estimation: y t = + ½y t 1 + ² t with Y where T T = T p; p = 1 1 Define: x t = (1 y t 1 ) with X =( ½) 0 Using least-squares: T 2 y t = x t + ² t! ^ =(X 0 X) 1 X 0 Y For example, if = 0 then: ^½ = ^ 1 ½ ^ 0
24 Consistency Define: u t = x 0 t² t with u t amartingaledi.sequence(mds)since E(u t jf t 1 )=E(x 0 t² t jf t 1 )=0 1 TX x 0 T t² t = 1 TX p u t! E(ut )=0 T t=p+1 t=p+1 1 TX By the ergodic theorem: x 0 p T tx t! E(x 0 t x t )=Q t=p+1 Ã! 1 Ã! 1 TX ^ = + x 0 1 TX T tx t x 0 p T t² t! Q 1 0=0 T t=p+1 T t=p+1
25 CLT for MDS If u t is a strictly stationary and ergodic MDS and E(u 0 tu t )=Ð< 1 then as T!1 1 TX d 1 TX d pt u t! N(0; Ð)! p x 0 t² t! N(0; Ð) T t=p+1 T t=p+1 Ð=E(x 0 tx t ² 2 t ) Hence, if y t is strictly stationary and ergodic with E(y t 4 ) < 1 then p d T ( ^ )! N(0;Q 1 ÐQ 1 )
26 AR(p) Models More generally: y t = + ½ 1 y t 1 + ::: + ½ p y t p + ² t Stationarity: (1 ½ 1 L ::: ½ p L p )y t = ² t From the fundamental theorem of algebra: ½(L) =(1 ½ 1 L :::½ p L p ) =) ½(z) =(1 1 1 z):::(1 1 p z) 1;:::; p are the roots (possibly some complex) such that ½( j) =0: Stationarity =) ) j jj j outside the unit circle
27 The Unit Circle Let j = a + bi
28 The VAR(1) representation of an AR(1) A convenient way to express an AR(p): y t = ½ 1 y t 1 + ::: + ½ p y t p + ² t yt ½1 ::: yt 1 ²t y 6 t = 6 ½ 1 ½ 2 ½ p 1 0 ::: y t ::: ::: ::: ::: y t p+1 0 ::: 1 0 y t p 0 Y t = BY t 1 + V t Stationarity: ti it all eigenvalues of B inside id the unit circle
29 Example Suppose: y t = y t 1 0:25y t 2 + ² t Therefore: 1 0:25 B = 1 0 with eigenvalues:» b = 1~ ~0;» eig(b); Notice: (1 0:5L)(1 0:5L) =1 L +0:25L 2
30 VAR(1) state-space t representation ti Also useful to generate forecasts since: Y t+h = B h Y t + V t+h + BV t+h 1 + ::: + B h 1 V t+1 Hence the forecast for y t+h we can just read off the first row from Y t+h Choosing the lag length p: Use Wald tests Use information criteria, e.g. AIC(p) =log^¾ 2 + 2p T
31 AR(p)! MA(1) If AR(p) is stationary, then the lag polynomial is invertible as an infinite MA process (actually, its Wold representation). A recursive formula: note (1 ½ 1 L ::: ½ p L p )y t =² t ½(L)y 1 t = ² t =) ½(L) ½(L)y t =(1 + à 1 L + :::)² t or (1 ½L ::: ½ p L p )(1 + à 1 + :::) =1 equating terms in L j à 1 L ½ 1 L =0 =) à 1 = ½ 1 à 2 L 2 ½ 2 L 2 ½ 1 à 1 L 2 =0 =) à 2 = ½ 1 à 1 + ½ 2 ::: à j = ½ 1 à j 1 + ½ 2 à j 2 + ::: + ½ j 1 à 1 + ½ j
32 Whyisthisuseful? Several reasons: 1. Recall that the variance of the forecast error of an AR(p) is: V (y t+h ^y t (h)) = (1 + Ã1 2 + Ã2 2 + ::: + Ãh 1)¾ To calculate impulse responses (sort of a treatment effect): IR(y t ; h) =E(y t+h j² t =1; F t 1 ) E(y t+h j² t =0; F t 1 ) =(0 + Ã 1 0+::: + Ã h 1+Ã h+1 0+:::) (0 + Ã 1 0+::: + Ã h 0+Ã h+1 0+:::) =Ã h
33 MA(q) Models The Wold representation theorem says that any covariance-stationary process has a (possibly) infinite MA representation. Now let me discuss MA(q) models briefly General specification: MA(1) y t = ¹ + ² t + μ 1 ² t 1 + ::: + μ q ² t q y t =¹ + ² t + μ² t 1 =¹ +(1+μL)² t
34 Properties of the MA(1) Expectation: E(y t )=E(¹ + ² t + μ² t 1 )=¹ Variance: Autocovariance: Autocorrelation: E(y t ¹) 2 = E(² t + μ² t 1 ) 2 =(1+μ 2 )¾ 2 E[(y t ¹)(y t j ¹)] = j =0forj>1 1 = E[(y t ¹)(y t 1 ¹) =E(² t + μ² t 1 )(² t 1 + μ² t 2 )] = μ¾ 2 Forecasts: ½ 1 = μ=(1 + μ 2 ); ½ j =0forj>1 ^y T (1) = ^¹ + ^μ^² T ;^y T (h) =^¹ for h>1 Forecast error variance: Impulse Response: E[(y T +h ^y T (h)) 2 ]=¾ 2 for h =1;(1+μ 2 )¾ 2 for h>1 IR(y t ; h) =μ for h =1;0otherwise
35 Estimation of MA models Usually by MLE (involving nonlinear numerical optimization routines). Assume: Then, conditional on y t j² t 1» N((¹ + μ² t 1 );¾ 2 ) ² 0 =0! ² 1 = y 1 ¹! ² 2 = y 2 ¹ μ(y 1 ¹)::: ² t = y t ¹ μ(y t 1 1 ¹ μ(y t 2 2 ¹ :::))) So that the likelihood of the t observation is f(y t j² t 1 ; ¹; μ) = 1 ( p exp (y t ¹ μ² t 1 ) 2 2¼¾ 2 2¾ 2
36 Log-LikelihoodLikelihood The log-likelihood therefore is: L(¹; μ) = T 2 log(2¼) T 2 log(¾2 ) TX t=1 ² 2 t 2¾ 2 A simpler way. Notice that if jμj < 1, i.e., the MA(1) is invertible, then y t = ¹ +(1+μL)² t =) y t (1 + μl) 1 = ¹ 1+μ + ² t y t (1 μl + μ 2 L 2 ::::) = ¹ + ² t 1 + μ y t = ¹ 1+μ + μy t 1 μ 2 y t 2 + ::: + ² t y t = c + 1y t 1 + 2y t 2 + ::: + ² t
37 Approximate Least-Squares MA(1) Thus: y ¹ 2 t = + μy t 1 μ y t 2 + ::: + ² t 1+μ Ory t = c + 1y t 1 + 2y t 2 + ::: + ² t So one approach is to estimate a truncated version since if jμj < 1thenμ j! 0;j!1 Choose a finite it p using, say AIC Regress by OLS y t = c + 1y t 1 + ::: + py t p + v t Obtain and hence Define And hence: (^ (c; 1; ^ :::; p) ^ ) = ¼ ^ ^Ð ¹ h(¹; μ) =( 1+μ ; μ; μ2 ; :::) g(^¼ g(¼; ¹; μ) = ¼ ^¼ h(¹; μ) Ð^¼
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