Springer Texts in Business and Economics

Time-Series Analysis

14.1 The AR(1) Model. yt = pyt_i + ©t with |p| <1 and ©t ~ IIN (0, a©2). Also, yo - N (0, o©2/1 - p2).

a. By successive substitution

yt = pyt_i + ©t = p(pyt_2 + ©t_1) + ©t = P2yt_2 + P©t_1 + ©t

= p2(pyt_3 + ©t_2) + p©t_1 + ©t = p3yt_3 + p2 ©t_2 + p©t_1 + ©t

= ••• = pVo c pt 1©i c pt 2©2 C ••• C ©t Then, E(yt) = ptE(yo) = 0 for every t, since E(yo) = E(©t) = 0.

var(yt) = p2tvar(yo) C p2(t 1)var(©i) C p2(t 2)var(©2) C------------------------------------ C var(©t)

If p = 1, then var(yt) = „©2/0 ! 1. Also, if |p| > 1, then 1 — p2 < 0 and var(yt) < 0.

b. The AR(1) series yt has zero mean and constant variance „2 = var(yt), for t = 0, 1 , 2, ... In part (a) we could have stopped the successive substitution at yt_s, this yields yt = psyt_s C pS 1©t_s+1 C • • C©t

Therefore, cov(yt, yt_s) = cov(psyt_s C Ps 1©t_s+1 C • • C©t, yt_s) = psvar(yt_s) = ps„2 which only depends on s the distance between t and t-s. Therefore, the AR(1) series yt is said to be covariance-stationary or weakly stationary.

B. H. Baltagi, Solutions Manual for Econometrics, Springer Texts in Business and Economics, DOI 10.1007/978-3-642-54548-1_14, © Springer-Verlag Berlin Heidelberg 2015

c. First one generates yo = 0.5 N(0,1)/(1 — p2)1/2 for various values of p. Then yt = pyt_i + et where et ~ IIN(0,0.25) for t = 1,2,.., T.

14.2 The MA(1) Model. yt = et + 0et_1 with et ~ IIN (0, о2)

a. E(yt) = E(et) + 0E(et_1) = 0 for all t. Also,

var(yt) = var(et) + 02var(et_1) = (1 + 02)oe2 for all t.

Therefore, the mean and variance are independent of t.

b. cov(yt, yt—1) = cov(et + 0St_1, St_1 + 0St_2) = 0var(8t_1) = 0oe2

0a©2 when s = 1 0 when s >1

0a©2 when s = 1 0 when s >1

for all t and s, then the MA(1) process is covariance stationary.

c. First one generates et ~ IIN(0,0.25). Then, for various values of 0, one generates yt = et + 0et_1.

14.3

a. The sample autocorrelation function for Income using EViews is as follows:

Correlogram of Income Sample: 1959 2007 Included observations: 49

Partial

Null Hypothesis: Y has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 0 (Automatic based on SIC, MAXLAG=10)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -1.843738 0.6677

Test critical values: 1% level -4.161144

5% level -3.506374

10% level -3.183002

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation Dependent Variable: D(Y)

Method: Least Squares

Sample (adjusted): 1960 2007 Included observations: 48 after adjustments

This does not reject the null hypothesis of unit root for Income. c. The Augmented Dickey-Fuller test statistic for differenced Income using a

constant and linear trend is as follows:

Null Hypothesis: D(Y) has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 0 (Automatic based on SIC, MAXLAG=10)

F-statistic 22.58242 Durbin-Watson stat 2.018360 Prob(F-statistic) 0.000000

This rejects the null hypothesis of unit root for differenced Income. We conclude that Income is I(1).

d. Let R1 denote the ols residuals from the regression of Consumption on Income and a constant. This tests R1 for unit roots: This ADF includes a constant

Null Hypothesis: R1 has a unit root Exogenous: Constant

Lag Length: 0 (Automatic based on SIC, MAXLAG=10)

‘MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation Dependent Variable: D(R1)

Method: Least Squares

Sample (adjusted): 1960 2007 Included observations: 48 after adjustments

Augmented Dickey-Fuller Test Equation Dependent Variable: D(R1)

Method: Least Squares

Both ADF tests do not reject the null hypothesis of unit roots in the ols residuals.

f. Correlogram of log(consumption)

Correlogram of log(consumption)

Sample: 1959 2007 Included Observations: 49

For log(consumption), the ADF with a Constant and Linear Trend yields:

Null Hypothesis: LOGC has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 1 (Automatic based on SIC, MAXLAG=10)

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation Dependent Variable: D(LOGC)

Method: Least Squares

Sample (adjusted): 1961 2007

Included observations: 47 after adjustments

Null Hypothesis: D(LOGC) has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 1 (Automatic based on SIC, MAXLAG=10)

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation Dependent Variable: D(LOGC,2)

Method: Least Squares

Sample (adjusted): 1962 2007

Included observations: 46 after adjustments

Correlogram of log Income Sample: 1959 2007 Included Observations: 49

Null Hypothesis: LOGY has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 0 (Automatic based on SIC, MAXLAG=10)

Augmented Dickey-Fuller Test Equation Dependent Variable: D(LOGY)

Method: Least Squares

We do not reject the null hypothesis that log(income) has unit root at the 5% level.

For differenced log(income), the ADF with a Constant and Linear Trend yields:

Null Hypothesis: D(LOGY) has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 0 (Automatic based on SIC, MAXLAG=10)

Augmented Dickey-Fuller Test Equation Dependent Variable: D(LOGY,2) Method: Least Squares

Sample (adjusted): 1961 2007 Included observations: 47 after adjustments

We do reject the null hypothesis that differenced log(income) has unit root at the 5% level.

The OLS regression of log(consumption) on log(income) and a constant yields:

Dependent Variable: LOGC Method: Least Squares

Sample: 1959 2007 Included observations: 49

Let R2 denote the ols residuals from the regression of Log(Consumption) on log(Income) and a constant.

This tests R2 for unit roots:

Null Hypothesis: R2 has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic based on SIC, MAXLAG=10)

Augmented Dickey-Fuller Test Equation Dependent Variable: D(R2)

Method: Least Squares

Sample (adjusted): 1960 2007 Included observations: 48 after adjustments

We do not reject the null hypothesis of unit roots in these ols residuals.

Johansen’s Cointegration Tests for Log(consumption) and log(Y)

Sample (adjusted): 1961 2007

Included observations: 47 after adjustments

Trend assumption: Linear deterministic trend

Series: LOGC LOGY

Lags interval (in first differences): 1 to 1

Unrestricted Cointegration Rank Test (Trace)

Hypothesized Trace

No. of CE(s) Eigenvalue Statistic

None 0.253794 14.26745

At most 1 0.010751 0.508018

Trace test indicates no cointegration at the 0.05 level * denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-values

Unrestricted Cointegration Rank Test (Maximum Eigenvalue)

Max-eigenvalue test indicates no cointegration at the 0.05 level * denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-values

Both tests indicate no cointegration between Log(consumption) and log(Y) at the 5% level

GARCH(1,1) for Log(consumption) and log(Y)

Dependent Variable: LOGC

Method: ML - ARCH (Marquardt) - Normal distribution

Sample: 1959 2007 Included observations: 49 Convergence achieved after 20 iterations Presample variance: backcast (parameter = 0.7) GARCH = C(3) + C(4)*RESID(-1)"2 + C(5)*GARCH(-1)

14.5 Data Description: This data is obtained from the Citibank data base.

Ml: is the seasonally adjusted monetary base. This a monthly average series. We get the quarterly average of M1 by using (M1t + M1t+i + M1t+2)/3. TBILL3: is the T-bill-3 month-rate. This is a monthly series. We

calculate a quarterly average of TBILL3 by using (TBILL3t + TBILL3t+1 + TBILL3t+2)/3. Note that TBILL3 is an annualized rate (per annum).

GNP: This is Quarterly GNP. All series are transformed by taking their natural logarithm.

a. VAR with two lags on each variable

Sample(adjusted): 1959:3 1995:2

Included observations: 144 after adjusting endpoints

Standard errors & t-statistics in parentheses

Determinant Residual Covariance 2.67E-11 Log Likelihood 1355.989

Akaike Information Criteria -24.24959

Schwarz Criteria -24.10523

b. VAR with three lags on each variable

Sample(adjusted): 1959:4 1995:2

Included observations: 143 after adjusting endpoints

Standard errors & t-statistics in parentheses

Determinant Residual Covariance 2.18E-11 Log Likelihood 1360.953

Akaike Information Criteria -24.40808

Schwarz Criteria -24.20088

d. Pairwise Granger Causality Tests Sample: 1959:1 1995:2 Lags: 3

Obs F-Statistic Probability

LNTBILL3 does not Granger Cause LNM1 143 20.0752 7.8E-11

LNM1 does not Granger Cause LNTBILL3 1.54595 0.20551

e. Pairwise Granger Causality Tests Pairwise Granger Causality Tests Sample: 1959:1 1995:2 Lags: 2

Obs F-Statistic Probability

LNTBILL3 does not Granger Cause LNM1 144 23.0844 2.2E-09

LNM1 does not Granger Cause LNTBILL3 3.99777 0.02051

14.6 The Simple Deterministic Time Trend Model. This is based on Hamilton (1994). yt = a + "t + ut t = 1,... ,T where ut ~ IIN(0, a2).

In vector form, this can be written as y = X® + u, X = [1,t], ® =

a. ®ols = (X'X)-1X'y and ®ols - ® = (X'X)-1X'u Therefore,

1 (T + 1)/2

(T + 1)/2 (T + 1)(2T + 1)/6

Therefore, plim(XjX) diverges as T! 1 and is not a positive definite matrix.

c. Note that

TT

TEi Т^Е t

t=i t=i

TT

T2Et ТзЕt2

L t=1 t=1 .

d. Show that z1 = - E ut ~ N(0, ct2). ut ~ N(0, ct2), so that Eut

VT t=i t=i

N(0,Tct2).

Therefore, —T P ut ~ N (0, - T • Tct2 • —t) = N(0, ct2).

T2

Also, show that z2 = t-t E tut ~ N 0, 6T. • (T + 1)(2T + 1) .

has an asymptotic distribution N(0, ct2Q). Hence, [A 1(X, X)A J] 1 [A-1(X0u)] has an asymptotic distribution N(0,Q-1ct2QQ-1) or N(0, o2Q-1). Thus,

T ((a ols - a)

TPT " ols - " ,

has an asymptotic distribution N(0, o2Q-1). Since "ols has the factor T/T rather than the usual VT, it is said to be superconsistent. This means that not only does ("ols — ") converge to zero in probability limits, but so does T("ols — "). Note that the normality assumption is not needed for this result. Using the central limit theorem, all that is needed is that ut is White noise with finite fourth moments, see Sims et al. (1990) or Hamilton (1994).

Test of Hypothesis with a Deterministic Time Trend Model. This is based on Hamilton (1994).

t

a. Show thatplim s2 = Tzy S(yt — 'ok — "olst)2 = o2. By the law of large

T

Hence, plim s2 = plimTij u2 = var(ut) = ct2

t=1

-y has the same asymptotic N(0,1)

2

s2(1,0)(X'X)-

distribution as t* = VT(a ois_a°). Multiplying both the numerator and

o qll

denominator of ta by VT gives

Q=

Now, [P'T, 0] in p can be rewritten as [1,0]A, because [1,0]A = [1,0] f TPT = [PT, 0].

Therefore, ta = ----------------- VT(Sols~a°)------------- r-

s2[1,0]A(X'X)-1A

(aols ao) , *

oVq^ " '

Л/Т (a ols ao)

_WT(Pols - "o)

totic distribution N(0, o2Q-1), so that pT (aols — ao) is asymptotically distributedN(0, o2q11). Thus,

, * Л/Т ('ols ao)

a-~^/qr~

is asymptotically distributed as N(0,1) under the null hypothesis of a = ao. Therefore, both ta and t* have the same asymptotic N(0,1) distribution. c. Similarly, fortesting Ho; " = "o, show that

t" = (" ols — "o)/[s2(0,1)(X0X)-1(0,1)0]1/2

has the same asymptotic N(0,1) distribution as t* = T/T(pols—p^/o^q22. Multiplying both numerator and denominator of tp by T/T we get,

tp = tVT(P ois — po)/[s2(0,TVT)(X'X)—1(0,tVT)']1/2

= tVT(|° ois — Po)/[s2(0,1)A(X, X)_1A(0,1)0]1/2

Now [0,TVT] in tp can be rewritten as [0,1]A because

Therefore, plim tp = TVT (pols — "o) /[o2(0,1)Q“1(0,1),]1/2 = TVT (pols — Po^ /oy^q22- Using plim s2 = o2 and plim A(X0X)_1A = Q_1, we get that plim [s2(0,1)A(X0X)_1A(0,1)0]1/2 = [o2(0,1)Q_1(0,1)0]1/2 = o^q22 where q22 is the (2,2) element of Q_1. Therefore, tp has the same asymptotic distribution as

TVT (pols — p^ /^q22 = tp*

From problem 14.6, part (e), T/T ols — po^ has an asymptotic distribu­tion N(0, o2q22). Therefore, both tp and t* have the same asymptotic N(0,1) distribution - Also, the usual OLS t-tests for a = ao and p = po will give asymptotically valid inference-

14.8 A Random Walk Model. This is based on Fuller (1976) and Hamilton (1994). yt = yt_1 + ut, t = 0,1,.., T where ut ~ IIN(0, o2) and yo = 0.

a. Show that yt = u1 h-------------- hut with E(yt) = 0 and var(yt) = to2. By successive

substitution, we get

yt = yt—1 + ut = yt—2 + ut-1 + ut = •• = yo + u1 + u2 h hut

substituting yo = 0 we get yt = u1 + •• +ut.

Hence, E(yt) = E(u1) + • • +E(ut) = 0

var(yt) = var(u1) + • • +var(ut) = to2

and yt - N(0, to2).

2 1

prj - 222 • T u2 is asymptotically distributed as 2 (x2 - 1).

c. Show that E ^P у2-^ = T(T2-1)o2. Using the results in part (a), we get

yt-1 = yo + u1 + u2 + •• +ut-1

Substituting yo = 0, squaring both sides and taking expected values, we get E (y2-i) = E (u2) C CE (u2-J = (t - 1)ct2 since the ut’s are independent.

Therefore,

E (E y?-t! = X E (yf-x) = E(t - V = T^T2L^-2

U=1

T

where we used the fact that t = T(T C 1)/2 from problem 14.6.

t=i

d. For the AR(1) model, yt = pyt-i C ut, show that OLS estimate of p satisfies

T

From part (b), T;? P yt-1ut has an asymptotic distribution 1 (x2 — 1).

° t=1

This implies that J2 yt-1ut/o2 converges to an asymptotic distribution of

2 (x? — 1) at the rate of T. Also, from part (c), E ^P y2-1^ = g T(T ^

T

t=1

to 2 at the rate of T2. One can see that the asymptotic distribution of p when p = 1 is a ratio of a 2 (xf — 1) random variable to a non-standard distribution in the denominator which is beyond the scope of this book, see Hamilton (1994) or Fuller (1976) for further details. The object of this exercise is to show that if p = 1, VT(p — p) is no longer Normal as in the standard stationary least squares regression with |p| < 1. Also, to show that for the non-stationary (random walk) model, p converges at a faster rate (T) than for the stationary case (VT). From part (c) it is clear that one has to divide the denominator of p by T2 rather than T to get a convergent distribution.

14.9 Cointegration Example

and

, -------------- Ut — ------------ V

(' - P) t (' - P)t (' - P)

In this case, ut is I(0) and vt is I(1). Therefore both Yt and Ct are I(1). Note

that there are no excluded exogenous variables in (14.13) and (14.14) and only one right hand side endogenous variable in each equation. Hence both equations are unidentified by the order condition of identification. However, a linear combination of the two structural equations will have a mongrel dis­turbance term that is neither AR(1) nor random walk. Hence, both equations are identified. If p = 1, then both u, and v, are random walks and the mon­grel disturbance is also a random walk. Therefore, the system is unidentified. In such a case, there is no cointegrating relationship between Ct and Yt. Let (с' - yY,) be another cointegrating relationship, then subtracting it from the first cointegrating relationship, one gets (y - P)Yt which should be I(0). Since Y' is I(1), this can only happen if y = P. Differencing both equations in (14.13) and (14.14) we get

Ac, - pAY' = Au, = (p - 1)u,-i + ©, = ©, + (p - 1)(C'_i - PY'_i)

= s, + (p - 1)C'-1 - P(p - 1)Y'-1

and AC, - aAY, = Av, = q,. Writing them as a VAR, we get (14.17)

-aet - a(p - 1)Ct_і + a"(p - 1)Yt_i +

—©t — (p — 1)Ct-i + "(p — 1)Yt-i + ht

where ht and gt are linear combinations of et and rp. This is Eq. (14.18). This

8 = (p — 1)/(" — a) and Zt = Ct — "Yt.

These are Eqs. (14.19) and (14.20). This is the Error-Correction Model (ECM) representation of the original model. Zt-1 is the error correction term. It represents a disequilibrium term showing the departure from long - run equilibrium. Note that if p = 1, then 8 = 0 and Zt-1 drops from both ECM equations.

T T

CtYt Ytut

b. "ols = " C tD1

T

Since ut is I(0) if p ф 1 and Yt is I(1), we have plim p Yt2/T2 is O(1),

t=1

T

while plimp; Ytut/T is O(1). Hence T("ols — ") is O(1) or ("ols — ") is O(T).

t=1

References

Fuller, W. A. (1976), Introduction to Statistical Time Series (John Wiley and Sons: New York).

Hamilton, J. D. (1994), Time Series Analysis (Princeton University Press: Princeton, New Jersey).

Sims, C. A., J. H. Stock and M. W. Watson (1990), “Inference in Linear Time Series Models with Some Unit Roots,” Econometrica, 58: 113-144.

P X'/o?] [POA)] - [p( Vo? r

b. From the regression equation Y, = a C "X, C u, one can multiply by w,*

and sum to get P w*Y, = a P w* C " P w*X, C P w*u,. Now divide

i= 1 i i= 1 i i= 1 i i= 1 i n

by w* and use the definitions of Y* and X* to get Y* = a C "X* C u*

i=1 i

nn

where u * = w*u, w*.

i=1 i i=1 i

[1]

P wi* (Xi — X*)2

i=1

_ 1 P wi* (Xi — X*)2

i=1

where the third equality uses the fact that the ui’s are not serially correlated and heteroskedastic and the fourth equality uses the fact that w* = (t/o2).