Econ 623 Econometrics II. Topic 3: Non-stationary Time Series. 1 Types of non-stationary time series models often found in economics

Transcription

1 Econ 623 Econometrics II opic 3: Non-stationary ime Series 1 ypes of non-stationary time series models often found in economics Deterministic trend (trend stationary): X t = f(t) + ε t, where t is the time trend and ε t represents a stationary error term (hence both the mean and the variance of ε t are constant. Usually we assume ε t ARMA(p, q) with mean 0 and variance σ 2 ) f(t) is a deterministic function of time and hence the model is called the deterministic trend model. Why non-stationary? E(X t ) = f(t)(depends on t, hence not a constant), V ar(x t ) = σ 2 (constant) Since ε t is stationary, X t is fluctuating around f(t), the deterministic trend. X t has no obvious tendency for the amplitude of the fluctuations to increase or decrease since V ar(x t ) is a constant. however.) (X t could increase or decrease, X t f(t) is stationary. For this reason a deterministic trend model is sometimes called a trend stationary model. Assume ε t = φ(l)e t, e t iid(0, σe). 2 If we label e t the innovation or the shock to the system, the innovation must have a transient, diminishing effect on X. Why? 1

2 Example: X t = δ 0 + δ 1 t + φ(l)e t, with φ(l) = 1 + αl + α 2 L 2 + and α < 1 So X t = δ 0 + δ 1 t + ε t, ε t = αε t 1 + e t ε t = φ(l)e t measures the deviation of the series from trend in period t. We wish to examine the effect of an innovation e t on ε t, ε t+1,... ε t = e t + αe t 1 + α 2 e t which gives ε t+s e t = α s (impulse response) herefore, when there is a shock to the economy, a deterministic trend model implies the shock has a transient effect. Sooner or later the system will be back to the deterministic trend. If f(t) = c 1 + c 2 t, we have a linear trend model. It is widely used. If f(t) = Ae rt, we have an exponential growth curve 2

3 If f(t) = c 1 + c 2 t + c 3 t 2, we have a quadratic trend model If f(t) = 1 k+ab t, we have a logistic curve 3

4 random walk Stochastic trend (unit root or difference stationarity): X t = µ + X t 1 + ε t where ε t is a stationary process (hence both the mean and the variance of ε t are constant. Usually we assume ε t ARMA(p, q) with mean 0 and variance σ 2 ) Another representation: (1 L)X t = µ + ε t Solving 1 L = 0 for L, we have L = 1, justifying the terminology unit root. Pure random walk: X t = X t 1 + e t, e t iid N(0, σ 2 e ) 1. X t = t 1 j=0 e t j if we assume X 0 = 0 with probability 1 2. E(X t ) = 0, V ar(x t ) = tσ 2 e 3. Non-stationarity. X t is wandering around and can be anywhere. 4

5 Random walk with a drift: X t = µ + X t 1 + e t, e t iid N(0, σ 2 e) 1. X t = tµ + t 1 j=0 e t j if we assume X 0 = 0 with probability 1 2. E(X t ) = tµ, V ar(x t ) = tσ 2 e 3. Non-stationarity. 4. As E(X t ) = tµ, a stochastic trend (or unit root) model could behave similar to a model with a linear deterministic trend. In a random walk model, X t = tµ+ t j=0 e t j. If we label e t j the innovation or the shock to the system, the innovation has a permanent effect on X t because X t e t j = 1, j > 0. his is true for all models with a unit root. herefore, when there is a shock to the economy, a unit root model implies that the shock has a permanent effect. he system will begin with a new level every time. In a trend-stationary model X t = δ 0 + δ 1 t + φx t 1 + e t, φ < 1, we have X t e t j = φ j 0. So the innovation has a transient effect on X t 5

6 A deterministic trend model and a stochastic trend (or unit root) model could behave similar to each other. However, knowing whether non-stationarity in the data is due to a deterministic trend or a stochastic trend (or unit root) would seem to be a very important question in economics. For example, macroeconomists are very interested in knowing whether economic recessions have permanent consequences for the level of future GDP, or instead represent temporary downturns with the lost output eventually made up during the recovery. If (1 L)X t = ε t ARMA(p, q), we say X t ARIMA(p, 1, q) or X t is an I(1) process. If (1 L) d X t = ε t ARMA(p, q), we say X t ARIMA(p, d, q) or X t is an I(d) process. 6

7 Why linear time trends and unit roots? Why linear trends? Indeed most GDP series, such as many economic and financial series seem to involve an exponential trend rather than a linear trend (apart from a stochastic trend). Suppose this is true, then we have Y t = exp(δt) However, if we take the natural log of this exponential trend function, we will have, ln Y t = δt hus, it is common to take logs of the data before attempting to describe them. After we take logs, most economic and financial time series exhibit a linear trend. his is why researchers often take the logarithmic transformation before doing analysis. Why unit roots? After taking logs, Suppose a time series follows a unit root rather than a linear trend, we have (1 L) ln Y t = ε t, where ε t is a stationary process, such as ARMA(p,q). However, (1 L) ln Y t = ln(y t /Y t 1 ) = ln(1 + Y t Y t 1 Y t 1 ) Yt Y t 1 Y t 1. So the rate of growth of the series Y t is stationary. For example, if Y t represents CPI and ln Y t is a I(1) process, then (1 L) ln Y t represent inflation rate and is a stationary process. 7

8 Explosive trend: X t+1 = φx t + ε t, φ > 1, X 0 = 0. Why explosive trends? Let P t be the stock price (or house price) at time t before the dividend payout or rental, D t be the dividend payoff from the asset at time t, and r be the discount rate (r > 0). he standard no arbitrage condition implies that and recursive substitution yields P t = r E t(p t+1 + D t+1 ), (1.1) where F t = (1 + r) i E t (D t+i ) and i=1 P t = F t + B t, (1.2) E t (B t+1 ) = (1 + r)b t. (1.3) the asset price is decomposed into two components, a fundamental component, F t, that is determined by expected future dividends, and a supplementary solution corresponding to the bubble component, B t. When B t = 0, P t = F t. Otherwise, P t = F t + B t and price embodies the explosive component B t. Consequently, under bubble conditions, P t will manifest the explosive behavior inherent in B t. Other forms of nonstationarity: structural break in mean, structural break in variance. 8

9 2 Processes with Deterministic rends Model with a simple linear trend X t = δt + e t, t = 1,...,. Let the OLS estimator of δ be δ, ie, δ = Xt t et t = δ +. t 2 t 2 If e t N(0, σ 2 ), then δ N ( δ, ) ( σ 2 = N δ, t 2 ) 6σ 2. ( + 1)(2 + 1) his is the exact small-sample normal distribution. he corresponding t statistic ( + 1)(2 + 1) δ 6 σ 2 has exact small-sample t distribution, where σ 2 = ( 1 X 1 t δ ) t. 9

10 If e t iid (0, σ 2 ) and has finite fourth moment, we have to find the asymptotic distributions for δ. Note that ) ( δ 3/2 δ = 1/2 e t t/. 3 t 2 Also note that {e t t/ } is a MDS with (1) σ 2 t = E(e 2 t t 2 / 2 ) = σ 2 t 2 / 2 and 1 σ 2 t 2 / 2 σ 2 /3; (2) E e t t/ 4 < ; (3) 1 e 2 t t 2 / 2 p σ 2 /3. he reason for (3) to hold is because E ( 1 e 2 t t 2 / 2 1 σ 2 t ) 2 = E ( 1 ( e 2 t σ 2) t 2 / 2 ) 2 = 1 6 E ( e 2 t σ 2) 2 t 4 0. Applying CL for MDS to {e t t/ }, we get 1/2 e t t/ d N(0, σ 2 /3). Since 3 t 2 1/3, 3/2 ( δ δ) d N(0, 3σ 2 ). Consider the t statistic: ( + 1)(2 + 1) ( δ δ) 6 σ 2 = ( + 1)(2 + 1) 3/2 ( δ δ) 6 σ 2 d N (0, 1).. 10

11 Model with a simple linear trend X t = α + δt + e t. If e t iid (0, σ 2 ) and has finite fourth moment, then [ 1/2 e t 1/2 (t/ ) e t ] ([ d 0 N 0 ] [ 1, σ ]). Need to find the asymptotic distributions for α and δ and the rate of convergence is different. [ 1/2 ( α α) ) ( δ 3/2 δ ] d N ( [ 0 0 ] [ 1, σ ] 1 ). he corresponding t statistics converge to N (0, 1). 11

12 3 Unit Root ests and A Stationarity est he theory of ARMA method relies on the assumption of stationarity. he assumption of stationarity is too strong for many macroeconomic time series. For example, many macroeconomic time series involve trend (both deterministic trends and stochastic trends unit root). ests for a unit root. Various test statistics are often used in practice: Augmented Dickey-Fuller (ADF) test, Phillips and Perron (PP) test. est for stationarity: Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test. hree possible alternative hypotheses for the unit root tests 1. no constant and no trend: X t = φx t 1 + e t, φ < 1 2. constant but no trend: X t = α + φx t 1 + e t, φ < 1 3. constant and trend: X t = α + φx t 1 + δt + e t, φ < 1 12

13 First consider the AR(1) process, X t = φx t 1 + e t, u t iid N(0, σ 2 ) Stationarity requires φ < 1 If φ = 1, X t is nonstationary and has a unit root est H 0 : φ = 1 against H 1 : φ < 1 How to estimate φ? OLS estimation, ie, φ = X tx t 1 X 2 t 1 = φ + X t 1 e t X 2. Stock t 1 (1994) shows that the OLS estimator of φ is superconsistent. What is the distribution or asymptotic distribution of φ φ? Recall from the standard model, y = Xβ + e If X is non-stochastic, β β N(0, σ 2 (X X) 1 ) If X is stochastic and stationary with some restrictions on autocovariance, ( β β) d N(0, σ 2 Q 1 ) where Q = p lim X X 13

14 3.1 Classical Asymptotic heory for Covariance Stationary Process Consider the AR(1) process, X t = φx t 1 + e t, e t iidn(0, σ 2 ) with φ < 1 For this model E(X t ) = 0, V ar(x t ) = What is the asymptotic property of X = 1 t=1 X t? We have E(X) = 0, V ar(x) = E(X 2 ) = V ar(xt) 0. So X r2 0 X p 0 σ2 σ 2 1 φ 2 X, γ j = φ j V ar(x t ). ( Also, V ar(x) j= γ j, which is called the long-run variance. φ φ ) φ he consistency is generally true for any covariance stationary process with condition j=0 γ j <. Martingale Difference Sequence (MDS): E(X t X t 1,..., X 1 ) = 0 for all t MDS is serially uncorrelated but not necessarily independent. MDS does not need to be stationary. 14

15 heorem (White, 1984): Let {X t } be a MDS with X = 1 t=1 X t Suppose that (a) E(X 2 t ) = σ 2 t > 0 with 1 t=1 σ2 t σ 2 > 0, (b) E X t r < for some r > 2 and all t, and (c) 1 t=1 X2 t p σ 2. hen X d N(0, σ 2 ). heorem (Andersen, 1971): Let X t = t=0 ψ jε t j where {ε t } is a sequence of iid variables with E(ε 2 t ) < and j=0 ψ j <. Let γ j be the j th order autocovariance. hen X d N(0, j= γ j) o know more about asymptotic theory for linear processes, read Phillips and Solo (1991). 15

16 3.2 Dickey-Fuller est Under H 0, however, we cannot claim ( φ φ) d N(0, σ 2 Q 1 ) because X t 1 is not stationary. If X t = φx t 1 + e t, and H 0 : φ = 1, we have φ 1 = X t 1 e t X 2 t 1 Consider X 2 t = (X t 1 + e t ) 2 = X t 1 e t = 1 2 (X2 t X 2 t 1 e 2 t ) = X t 1 e t = 1 2 (X2 X2 0 e 2 t ) Let X 0 = 0, X t 1 e t = 1 2 (X2 e 2 t ) e t iid (0, σ 2 ) = e 2 t iid with E(e 2 t ) = σ 2. By LLN, 1 e 2 t p σ 2 X = X 0 + t=1 e t = e t a N(0, σ 2 ) = X σ d N(0, 1) = X2 σ 2 d χ 2 (1) 1 σ 2 Xt 1 e t = 1 ( X2 e 2 t ) d 1 2 σ 2 σ 2 2 (χ2 (1) 1) What is the limit behavior of X 2 t 1? We need to introduce the Brownian Motion. 16

17 3.2.1 Brownian Motion Consider X t = X t 1 + e t, X 0 = 0, e t iidn(0, 1) = X t = t j=1 e j N(0, t) t > s, X t X s = t j=1 e j s j=1 e j = e s e t N(0, (t s)) Also note that, X t X s is indept of X r X q if t > s > r > q What happen if we go from the discrete case to the continuous case? We have the standard Brownian Motion (BM). It is denoted by B(t) A stochastic process B t over time is a standard Brownian motion if for small time interval, the change in B t, B t ( B t+ B t ), follows the following properties: 1. B t = e where e N(0, 1) 2. B t and B s for changes over any non-overlapping short intervals are independent he above two properties imply the following: 1. E( B t ) = 0 2. Var( B t ) = 3. σ( B t ) = 17

18 In general we have Due to the above, we have B B 0 = N e i i=1 E(B B 0 ) = 0 Var(B B 0 ) = σ(b B 0 ) = 18

19 herefore, for the Brownian motion, results for the mean and variance in small intervals also apply to large intervals. When 0, we write the Brownian motion as: db t = e where db t should be interpreted as the change in the Brownian motion over an arbitrarily small time interval. 19

20 Sample paths of the Brownian motion are continuous everywhere but not smooth anywhere. Derivative of the Brownian motion does not exist anywhere and the process is infinitely kinky. A Brownian motion is nonstationary as Var(B ) drifts up with t. he transition density is B + B N(B, ). A Brownian motion is also called a Wiener process. It has no drift. Now we extend it to the generalized Wiener process. We can define e t = e 1t + e 2t, e 1t, e 2t N(0, 1/2), y t+ 1 2 Definition of the BM: W (t) = σb(t) = y t + e 1t 20

21 Note: 1. B(0) = 0 2. B(t) N(0, t) or B(t) is a Gaussian process 3. E(B(t + h) B(t)) = 0, E(B(t + h) B(t)) 2 = h 4. E(B(t)B(s)) = min{t, s} 5. If t > s > r > q, B(t) B(s) is indept of B(r) B(q), ie, B(t) has indept increment 6. Since B(t) is a Gaussian process, it is completely specified by its covariance matrix 7. If B(t) is a BM, then so is B(t + a) B(a), λ 1 B(λ 2 t), B(t) 8. he sample path of BM is continuous in time with probability 1 9. No point differentiable 21

22 Return to the unit root test. X t = X 0 + t j=1 e j. Define p t = t j=1 e j = p t 1 + e t Change the time index from = {1,..., } to t with 0 t 1 Define Y (t) = 1 σ p [ t], if j 1 For instance, Y (1) = 1 σ p 0 if p 1 Y (t) = if 0 t 1 σ p j 1 σ if 1 t 2 j 1 wo important theorems t j t j, j = 1,...,. [ t] = integer part of t 1. Functional CL: Y ( ) d B( ) 2. Continuous mapping theorem: if f is a continuous function, f(y (t)) f(b(t)) d 22

23 Note that Y (1) d N(0, 1), Y (1/2) d N(0, 1/2), Y (r) d N(0, r), r [0, 1]. r 1, r 2 [0, 1], r 2 > r 1, Y ( r 2 ) Y ( r 2 ) d N(0, r 2 r 1 ). t=1 X t = t=1 p t 1 + t=1 e t if X 0 = 0. j/ Y (j 1)/ (t)dt = p j 1 σ = p j 1 = σ j/ j=1 Y (j 1)/ (t)dt = σ 1 Y 0 (t)dt t=1 X t = σ 1 Y 0 (t)dt + ej = t=1 Xt = σ 1 Y 0 (t)dt + 1 ej By FCL and CM, t=1 Xt d σ 1 0 B(t)dt 23

24 t=1 X2 t 1 = X X 2 1 = X X 2 1 = t=1 X2 t X 2 = t=1 p2 t p 2 Recall p 2 j 1 σ 2 Y 2 (1) = σ 2 2 j/ j 1/ Y 2 (t)dt = p 2 j 1 = σ Y 2 (t)dt and p2 = herefore, t=1 X2 t 1 2 d σ B2 (t)dt Under H 0, ( φ φ) d 1/2(χ2 (1) 1) 1 0 B2 (t)dt Phillips (1987) proved the above result under very general conditions. Note: 1. his limiting distribution is non-standard 2. he numerator and denominator are not independent. 3. It is called the Dickey-Fuller distribution since Dickey and Fuller (1976) uses Monte-Carlo simulation to find the critical values of them. χ 2 (1) B2 (t)dt and tabulate 24

25 1. 1/2 t=1 e d t σb(1); t=1 e t O p ( 1/2 ) 2. 1 t=1 p t 1e t d 1 2 σ2 {B 2 (1) 1} ; t=1 p t 1e t O p ( ) 3. 3/2 t=1 te d t σb(1) σ 1 W (r)dr; 0 t=1 te t O p ( 3/2 ) 4. 3/2 t=1 p d t 1 σ 1 B(r)dr; 0 t=1 p t 1 O p ( 3/2 ) 5. 2 t=1 p2 d t 1 σ B2 (r)dr; t=1 p2 t 1 O p ( 2 ) 6. 5/2 t=1 tp d t 1 σ 1 rb(r)dr; 0 t=1 tp t 1 O p ( 5/2 ) 7. 3 t=1 tp2 d t 1 σ rb2 (r)dr; t=1 tp2 t 1 O p ( 3 ) 8. t=1 t = ( + 1)/2 O( 2 ) 25

26 3.2.2 Case 1 (no drift or trend in the regression): In case 1, we test { H0 : X t = X t 1 + e t H 1 : X t = φx t 1 + e t with φ < 1 his is equivalent to { H0 : X t = e t H 1 : X t = βx t 1 + e t with β < 0 We estimate the following model using the OLS method: X t = βx t 1 + e t, e t iid (0, σ 2 e). 26

27 he OLS estimator of β is defined by β, which is biased but consistent. he test statistic (known as the DF Z test or the coefficient statistic) is defined as β. Under H 0, β d 1/2(χ2 1) (1). 1 0 B2 (r)dr 1/2(χ2 (1) 1) 1 0 B2 (r)dr is not a normal distribution. See able B.5 for the critical values of this distribution. he finite sample distribution of β B.5 for the critical values for different sample sizes. is not normal. See able Alternatively, we can use t-statistic β ŝe( β ) = β ( 2 Xt 1) 2 1/2 σ β ŝe( β ). Under H 0, it can be shown that d ) 1/2 (χ 2(1) 1 [ ] 1/ B2 (r)dr Although it is called the Dickey-Fuller t-statistic, it is no longer a t distribution. See able B.6 for the critical values of this distribution and the critical values of the finite sample distribution for different sample sizes. 27

28 3.2.3 Case 2 (constant but no trend in the regression): We test { H0 : X t = e t H 1 : X t = α + βx t 1 + e t with β < 0 In case 2, we estimate the following model using the OLS method: X t = α + βx t 1 + e t, e t iid N(0, σ 2 e) he DF Z test statistic is β. Under H 0, β d 1/2(χ2 (1) 1) B(1) 1 0 B(r)dr 1 0 B2 (r)dr ( 1 0 B(r)dr)2. See able B.5 for the critical values of this distribution and the critical values of the finite sample distribution for different sample sizes. Proof of the Asymptotic Distribution: he OLS estimates are [ ] [ α = β Xt 1 ] 1 [ ] et Xt 1 X 2 = t 1 Xt 1 e t [ Op ( ) O p ( 3/2 ) O p ( 3/2 ) O p ( 2 ) ] 1 [ Op ( 1/2 ) O p ( ) ]. 28

29 Let g( ) = diag( 1/2, ), then [ 1/2 α β ] [ = g( ) = { [ g 1 ( ) ] 1 [ ] Xt 1 et Xt 1 X 2 t 1 Xt 1 e t ] 1 [ ] Xt 1 Xt 1 X 2 g 1 ( )} g 1 et ( ) t 1 Xyt 1 e t [ 1 ] = 3/2 1 [ X t 1 3/2 X t 1 2 Xt 1 2 [ d 1 σ ] 1 1 B(r)dr [ 0 σ 1 B(r)dr 0 σ2 1 0 B2 (r)dr β ŝe( β ) 1/2 e t 1 X t 1 e t ] σb(1) 1 2 σ2 {B 2 (1) 1} d 1/2(χ2 (1) 1) B(1) 1 0 B(r)dr ( 1 0 B2 (r)dr ( 1 Alternatively, we can use t-statistic 0 B(r)dr)2) 1/2 under H 0. Although it is called the Dickey-Fuller t-statistic, it is no longer a t distribution. See able B.6 for the critical values of this distribution and the critical values of the finite sample distribution for different sample sizes. ]. 29

30 3.2.4 Case 2b (H 0 : Random Walk with Drift): What if the true model is a random walk with drift, ie, X t = α + X t 1 + e t = X 0 + tα + t j=1 e j? (α 0) X t 1 = X 0 + ( 1)α/2 + p t 1 O p ( 2 ) 2 X t 1 p α/2, 3 X 2 t 1 p α 2 /3, X t 1 e t O p ( 3/2 ) Let g( ) = diag( 1/2, 3/2 ), then [ 1/2 α 3/2 β ] = { [ g 1 ( ) ] } 1 [ Xt 1 Xt 1 X 2 g 1 ( ) t 1 1/2 e t 3/2 X t 1 e t ] [ ] 1 1 α/2 he first term converges to α/2 α 2. Since /3 3/2 p X t 1e t 3/2 α(t 1)e t, the second term converges to [ 1/2 e t 3/2 X t 1 e t ] ([ d 0 N 0 ], σ 2 [ 1 α/2 α/2 α 2 /3 ]). And hence, [ 1/2 α 3/2 β ] d N ( [ 0 0 ], σ 2 [ 1 α/2 α/2 α 2 /3 ] 1 ) 30

31 3.2.5 Case 3 (constant and trend in the regression): We test { H0 : X t = e t H 1 : X t = α + βx t 1 + δt + e t with β < 0 or { H0 : X t = α + e t H 1 : X t = α + βx t 1 + δt + e t with β < 0 In this case, we estimate the following model using the OLS method: X t = α + φx t 1 + δt + e t, e t iid N(0, σ 2 e) X t = α(1 φ) + φ(x t 1 α(t 1)) + (δ + φα)t + e t = α + φp t 1 + δ t + e t, where p t 1 = X t 1 α(t 1) is a pure random walk if φ = 1 (ie β = 0) and δ = 0. 31

32 Let g( ) = diag( 1/2,, 3/2 ), then 1/2 α 1 ( φ 1) ) = pt 1 t 1/2 e t ( δ g 1 ( ) pt 1 p 2 t 1 tpt 1 g 1 ( ) 1 p t 1 e t 3/2 α t tpt 1 t 2 3/2 te t 1 σ Bdr 1/2 d σ B(r)dr σ 2 B 2 dr σ 1 σb(1) rbdr 1/2 σ 1 2 σ2 [B 2 (1) 1] rbdr 1/3 σ [ B(1) Bdr ] 1 σ Bdr 1/2 B(1) = Bdr B 2 dr rbdr 1 2 [B2 (1) 1] 0 0 σ 1/2 rbdr 1/3 B(1). Bdr So the aymptotic distribution of φ (and hence β ) is independent of σ and α. 32

33 he Dickey-Fuller Z test statistic is β. Under H 0, the asymptotic distribution is not a normal distribution. See able B.5 for the critical values of the asymptotic distribution distribution and the critical values of the finite sample distribution for different sample sizes. Alternatively, we can use the Dickey-Fuller t-statistic distribution distribution is no longer a t distribution. β ŝe( β ) critical values of this distribution for different sample sizes. est Statistic 1% 2.5% 5% 10% Case Case Case whose asymptotic See able B.6 for the When the sample size is 100, the critical values for the Dickey-Fuller t-statistic are given in the table below est Statistic 1% 2.5% 5% 10% Case Case Case

34 3.2.6 Which case to use in practice? Use Case 3 test for a series with an obvious trend, such as GDP Use Case 2 test for a series without an obvious trend, such as interest rates. For variables such as interest rate we should use Case 2 rather than Case 1 since if the data were to be described by a stationary process, surely the process would have a positive mean. However, if you strongly believe a series has a zero mean when it is stationary, use Case 1. 34

35 3.3 Augmented DF (ADF) est he model in the null hypothesis considered in the DF test is a class of highly restrictive unit root models. Suppose the model in the null hypothesis is X t = X t 1 + u t, u t = κu t 1 + e t. Unless κ = 0, the DF test is not applicable. he above model implies the following AR(2) model for X t, X t = (1 + κ)x t 1 κx t 2 + e t = X t 1 + κ X t 1 + e t In general, if X t = φ 1 X t 1 + φ 2 X t φ p X t p + e t, e t iid(0, σ 2 e), (3.4) then the DF test is not applicable. Model (3.4) can be represented by X t = φx t 1 + φ 1 X t φ p 1 X t p+1 + e t, e t iid(0, σ 2 e), where φ = p i=1 φ i So if φ = 1, {X t } is a unit root process; if φ < 1, {X t } is a stationary process. 35

36 In Case 2, the ADF test is based on the OLS estimator of β from the following regression model, X t = α + βx t 1 + φ 1 X t φ p 1 X t p+1 + e t, e t iid(0, σ 2 e) In Case 3, the ADF test is based on the OLS estimator of β from the following regression model, X t = α + δt + βx t 1 + φ 1 X t φ p 1 X t p+1 + e t, e t iid(0, σ 2 e) Based on the OLS estimator of β, the ADF test statistic, has the same asymptotic distribution as the DF t statistic. β, can be used. It ŝe( β ) 36

37 3.4 Phillips-Perron (PP) est X t = X t 1 + u t, Φ(L)u t = Θ(L)e t, e t iidn(0, σe), 2 so u t is an ARMA process. Phillips and Perron (1988) proposed a nonparametric method of controlling for possible serial correlation in u t. here are two PP test statistics. One of them (so-called PP Z test) is the analogue of β. he other (so-called PP t test) is the analogue of Z test has the same sampling distribution as β previous section. PP t test has the same sampling distribution as three cases in the previous section. β ŝe( β ). PP under all three cases in the β ŝe( β ) under all In Case 2, both PP tests are based on the OLS estimator of β from the following regression model, X t = α + βx t 1 + ε t, where ε t ARMA In this case, the PP Z test statistic is defined by β 1 2 ( 2 σ 2 β s 2 )(λ2 γ 0 ), where s 2 = ( 2) 1 (X t α φ X t 1 ) 2, λ = σθ(1), γ 0 = E(u 2 t ) In Case 3, both PP tests are based on the OLS estimator of β from the following regression model, X t = α + δt + βx t 1 + ε t, where ε t ARMA 37

38 3.5 Kwiatkowski, Phillips, Schmidt, and Shin (KPSS, 1992) est Unlike the tests we have thus far introduced, KPSS test is a test for stationarity or trend-stationarity, that is, it tests for the null of stationarity or trend stationarity against the alternative of a unit root. It is assumed that one can decompose X t into the sum of a deterministic trend, a random walk and a stationary error X t = ξt + r t + u t (3.5) r t = r t 1 + e t (3.6) where e t iid N(0, σ 2 e). he initial value r 0 is assumed to be fixed. If σ 2 e = 0, X t is trend-stationary. If σ 2 e > 0, X t is a non-stationary. 38

39 KPSS test is a one-sided Lagrange Multiplier (LM) test he KPSS statistic is based on the residuals from the OLS regression of X t on the exogenous variables Z t = 1 or (1, t): X t = δ Z t + ɛ t he LM statistic is be defined as: t=1 S(t)2 2 f 0, where f 0 is an estimator of the long-run variance and S(t) is a cumulative residual function: S(t) = t ˆɛ s s=1 Critical values for the KPSS test statistic are: Level of Significance 10% 5% 2.5% 1% Crit value (case 2) Crit value (case 3)

40 If the LM test statistic is larger than the critical value, we have to reject the null of stationarity (or trend-stationarity). In general ɛ t is not a white noise. As a result, the long run variance is different from the short run variance. One way to estimate the long run variance is to nonparametrically estimate the spectral density at frequency zero, that is, compute a weighted sum of the autocovariances, with the weights being defined by a kernel function. For example, a nonparametric estimate based on Bartlett kernel is ˆf 0 = 1 h= ( 1) ˆγ(h)K(h/l), where l is a bandwidth parameter (which acts as a truncation lag in the covariance weighting), and ˆγ(h) is the h-th sample autocovariance of the residuals ˆɛ, defined as, ˆγ(h) = t=h+1 ˆɛ tˆɛ t h /, and K is the Barlett kernel function defined by K(x) = { 1 x if x 1 0 otherwise 40

41 4 Revisit to Box-Jenkins If the null hypothesis cannot be rejected, difference the data (ie, X t X t 1 = u t ) to achieve stationarity. In the case where both a stochastic trend and a linear deterministic trend are involved, the first difference transformation leads to a stationary process. est for a unit root in the differenced data. If the null hypothesis cannot be rejected, difference the differenced data until stationarity is achieved. ARMA(p, q)+one unit root =ARIMA(p, 1, q) ARMA(p, q)+d unit roots =ARIMA(p, d, q) 5 Limitations of Box-Jenkins he Data Generating Process has to be time-invariant. his assumption can be too restrictive for an economy undergoing a period of transition and for data covering a long sample period. What if the data is explosive, ie φ > 1. What if a nonlinear deterministic trend is involved. What if the data follow a trend stationary process. Unit root tests and model selection are done in two separate steps. 41

42 6 Explosive Models Model: X t+1 = φx t + ε t, φ > 1, X 0 = 0. Let φ be the LS estimator of φ. We will show that when ε t N(0, σ 2 ), φ φ 2 1 ) ( φ φ Cauchy. (6.7) When ε t iid (0, σ 2 ) but is not necessarily normally distributed, then φ φ 2 1 ) ( φ φ P/Q, where P and Q are the limits of P and Q defined by P = o prove (6.7), note that φ t ε t and Q = t=1 φ ( t) ε t. (6.8) t=1 X t = ε t + φε t φ t 1 ε 1 + φ t X 0 = ε t + φε t φ t 1 ε 1. Let P t = X t /φ t = φ t ε t + φ t+1 ε t φ 1 ε 1 which is normally distributed. P t is a martingale since E(P t I t 1 ) = φ t+1 ε t φ 1 ε 1 = P t 1 and E(Pt 2 ) = t s=1 φ 2s σ 2 = t s=1 σ2 (1 φ 2(t 1) )/(1 φ t ), and hence sup t E(Pt 2 ) = s=1 σ2 (1 φ 2(t 1) )/(1 φ 2 ) = σ 2 /(φ 2 1). 42

43 By the Martingale Convergence heorem, P t a.s. P = φ 1 ε φ t ε t +... N(0, σ 2 /(φ 2 1)). Let L be an integer and satisfies 1 L + L 0 as. Consider 1 φ 2 t=1 X 2 t 1 = 1 φ 2 = 1 φ 2 = 1 φ 2 = 1 φ 2 t=1 t=l+1 t=l+1 t=1 ( Xt 1 φ t 1 ( Xt 1 ) 2 φ 2(t 1) ) 2 φ 2(t 1) + 1 L ( ) 2 Xt 1 φ 2(t 1) φ t 1 φ 2 φ t 1 t=1 ( ) Lφ (P + o a.s. (1)) 2 φ 2(t 1) 2L + O a.s. φ 2 P 2 φ 2(t 1) + o a.s. (1) = 1 φ P 2 φ2 1 2 φ 2 1 a.s. P 2 φ

44 Consider 1 φ t=1 X t 1 ε t = 1 φ = 1 φ = 1 φ = 1 φ t=1 t=l+1 t=l+1 t=1 ( Xt 1 φ t 1 ( Xt 1 ) ε t φ t 1 ) ε φ t 1 t φ t L ( ) Xt 1 (εt φ t 1) φ φ t 1 t=1 ( ) Lφ (P + o a.s. (1)) ε t φ t 1 L + O a.s. P ε t φ t 1 + o a.s. (1) = P 1 φ φ ε t φ t 1 + o a.s. (1) t=1 = P [ φ 1 ε t + φ 2 ε t φ ε 1 ] + oa.s. (1). Q t = φ 1 ε t + φ 2 ε t φ t ε 1 is normally distributed and Q t a.s. Q N(0, σ 2 /(φ 2 1)). Note that Cov(P, Q ) = t=1 σ2 /φ +1 = σ 2 /φ So P and Q are asymptotically independent, and P and Q are independent. So φ φ 2 1 ( φ φ ) = 1 φ φ 1 φ 2 t=1 X t 1ε t t=1 X2 t 1 a.s. P Q P 2 = Q P = Cauchy. 44

45 Asymptotic distribution of OLS estimator of AR coefficient Model Asymptotic distribution ) X t = φx t + ε t, φ < 1, X 0 O p (1) ( φ φ N(0, 1 φ 2 ) ) X t = d + φx t + ε t, φ < 1, X 0 O p (1) ( φ φ N(0, 1 φ 2 ) ) X t = φx t + ε t, φ = 1, X 0 O p (1) ( φ 1 BdB/ B 2 ) BdB B(1) B X t = d + φx t + ε t, φ = 1, d = 0, X 0 O p (1) ( φ 1 ) B 2 ( [ B] 2 ) X t = d + φx t + ε t, φ = 1, d 0, X 0 O p (1) ( φ 3/2 1 N 0, 12σ2 d 2 X t = φx t + ε t, X 0 = 0, φ > 1, ε t iid N(0, σ 2 ) φ φ 2 1 ) ( φ φ Cauchy 45