Chaper 16 Regression wih Time Series Daa The analysis of ime series daa is of vial ineres o many groups, such as macroeconomiss sudying he behavior of naional and inernaional economies, finance economiss who sudy he sock marke, agriculural economiss who wan o predic supplies and demands for agriculural producs. We inroduced he problem of auocorrelaed errors when using ime series daa in chaper 12. In chaper 15 we considered disribued lag models. In boh of hese chapers we made implici saionary assumpions abou he ime series daa. In he conex of he AR(1) model of auocorrelaion, e = ρ e 1 + v, we assumed ha ρ < 1. In he infinie geomeric lag model, assumed φ< 1. y =α+ β x + e, where i i i= 1 i β i =βφ, we Slide 16.1
These assumpions ensure ha he ime series variables in quesion are saionary ime series. However, many of he variables sudied in macroeconomics, moneary economics and finance are nonsaionary ime series. The economeric consequences of nonsaionariy can be quie severe, leading o leas squares esimaors, es saisics and predicors ha are unreliable. Moreover, he sudy of nonsaionary ime series is one of he fascinaing recen developmens in economerics. In his chaper we examine hese and relaed issues. Slide 16.2
16.1 Saionary Time Series Le y be an economic variable ha we observe over ime. Examples of such variables are ineres raes, he inflaion rae, he gross domesic produc, disposable income, ec. The variable y is random, since we can no perfecly predic i. We never know he values of hese variables unil hey are observed. The economic model generaing y is called a sochasic or random process. We observe a sample of y values, which is called a paricular realizaion of he sochasic process. I is one of many possible pahs ha he sochasic process could have aken. The usual properies of he leas squares esimaor in a regression using ime series daa depend on he assumpion ha he ime series variables involved are saionary sochasic processes. A sochasic process (ime series) y is saionary if is mean and variance are consan over ime, and he covariance beween wo values from he series depends only on he Slide 16.3
lengh of ime separaing he wo values, and no on he acual imes a which he variables are observed. In oher words, he ime series y is saionary if for all values i is rue ha E( y ) = µ [consan mean] (16.1.1a) var ( ) 2 y = σ [consan variance] (16.1.1b) ( y y ) ( y y ) cov, = cov, =γ [covariance depends on s, no ] (16.1.1c) + s s s In Figure 16.1 (a)-(b) we plo some arificially generaed, saionary ime series. Noe ha he series vary randomly a a consan level (mean) and wih consan dispersion (variance). In Figure 16.1 (c)-(d) are plos of series ha are no saionary. These ime series are called random walks, because hey slowly wander upwards or downwards, bu wih no real paern. Slide 16.4
In Figure 16.1 (e)-(f) are wo more nonsaionary series, bu hese show a definie rend eiher upwards or downwards. These are called random walks wih a drif. The series in Figure 16.1 are generaed from an AR(1) process, much like he AR(1) error process we discussed in Chaper 12. The AR(1) process we consider is AR(1) process y = α+ρ y 1 + v (16.1.2) The AR(1) process is saionary if ρ < 1, as is he case in Figure 16.1 (a)-(b). If α = 0 and ρ = 1 he AR(1) process reduces o a nonsaionary random walk series, depiced in Figure 16.1 (c)-(d), in which he value of y his period is equal o he value y 1 from he previous period plus a disurbance v. Random Walk y = y 1 + v (16.1.3) Slide 16.5
A random walk series shows no definie rend, and slowly urns one way or he oher. If α 0 and ρ = 1 he series produced is also nonsaionary and is called a random walk wih a drif. Random Walk wih drif y = α+ y 1 + v (16.1.4) Such series do show a rend, as illusraed in Figure 16.1 (e)-(f). Many macroeconomic and financial ime series are nonsaionary. In Figure 16.2 we plo ime series of some imporan economic variables. Compare hese plos o hose in Figure 16.1. Which ones look saionary? The abiliy o disinguish saionary series from nonsaionary series is imporan because, as we noed earlier, using nonsaionary variables in regression can lead o leas squares esimaors, es saisics and predicors ha are unreliable and misleading, as we illusrae in he nex secion. Slide 16.6
16.2 Spurious Regressions There is a danger of obaining apparenly significan regression resuls from unrelaed daa when using nonsaionary series in regression analysis. Such regressions are said o be spurious. To illusrae he problem, le us ake he random walk daa from Figure 16.1 (c)-(d) and esimae a regression of series one (y = rw1) on series wo (x = rw2). These series were generaed independenly and have no relaion o one anoher. Ye, when we plo hem, Figure 16.3, we see an inverse relaionship beween hem. If we esimae he simple regression we obain he resuls in Table 16.1. These resuls indicae ha he simple regression model fis he daa well (R 2 =.75), and ha he esimaed slope is significanly differen from zero ( = 54.67). These resuls are compleely meaningless, or spurious. The apparen significance of he relaionship is false, resuling from he fac ha we have relaed one slowly urning series o anoher. Slide 16.7
Similar and more dramaic resuls are obained when he random walk wih drif series are used in a regression. Noe ha he Durbin-Wason saisic is low. Table 16.1 Spurious regression resuls Reg Rsq 0.7495 Durbin-Wason 0.0305 Variable DF B Value Sd Error Raio Approx Prob Inercep 1 14.204040 0.5429 26.162 0.0001 RW2 1-0.526263 0.00963-54.667 0.0001 Granger and Newbold sugges ha a Rule of humb is ha when esimaing regressions wih ime series daa, if he R 2 value is greaer han he Durbin-Wason saisic, hen one should suspec a spurious regression. To summarize, when nonsaionary ime series are used in a regression model he resuls may spuriously indicae a significan relaionship when here is none. In hese Slide 16.8
cases he leas squares esimaor and leas squares predicor do no have heir usual properies, and -saisics are no reliable. Since many macroeconomic ime series are nonsaionary, i is very imporan ha we ake care when esimaing regressions wih macro-variables. Slide 16.9
16.3 Checking Saionariy Using he Auocorrelaion Funcion In Equaion (16.1.1c) we defined he covariance beween y and definiion we can consruc he auocorrelaion funcion, ρ s, of he series as y + s. Using his ( y y+ s) γ s ( y ) 0 cov, ρ s = = (16.3.1) var γ The value of ρ 0 = 1, and for s > 1 he correlaions ρ s are pure numbers (uniless) beween 1 and 1. The esimaed sample correlaions are ( y y+ s) γ s? ( y ) 0 cov ˆ, ˆ ρ ˆ s = = (16.3.2) var γ Slide 16.10
where he sample variance and covariance are esimaed from a sample of size T as γ ˆ = s γ ˆ = 0 ( y y)( y y) + s ( y y) 2 T T (16.3.3) If we plo he sample correlaions ρ ˆ s agains s we obain wha is called a correlogram. Economeric sofware will compue he sample correlaions. In Tables 16.2 and 16.3 we show he firs 10 correlaions (AC) for he saionary series s2 and he nonsaionary series rw1. For he saionary series he auocorrelaions, he column labeled AC in Table 16.2, gradually die ou, indicaing ha values furher in he pas are less correlaed wih he curren value. Slide 16.11
Table 16.2 Correlogram for s2 Auocorrelaion s AC Q-Sa Prob. ******* 1 0.900 813.42 0.000. ****** 2 0.803 1461.0 0.000. ****** 3 0.718 1979.1 0.000. ***** 4 0.629 2377.9 0.000. **** 5 0.545 2677.4 0.000. **** 6 0.470 2900.7 0.000. *** 7 0.408 3068.7 0.000. *** 8 0.348 3191.2 0.000. ** 9 0.299 3281.8 0.000. ** 10 0.266 3353.2 0.000 Table 16.3 Correlogram for rw1 Auocorrelaion s AC Q-Sa Prob. ******** 1 0.997 997.31 0.000. ******** 2 0.993 1988.8 0.000. ******** 3 0.990 2973.9 0.000. ******** 4 0.986 3953.2 0.000. ******** 5 0.983 4926.3 0.000. ******** 6 0.979 5893.4 0.000. ******** 7 0.975 6854.4 0.000. ******* 8 0.972 7809.4 0.000. ******* 9 0.968 8758.3 0.000. ******* 10 0.965 9701.0 0.000 Slide 16.12
For he nonsaionary series rw1, he auocorrelaions in Table 16.3 do no die ou rapidly a all. The correlaion beween rw1 and rw1-10 is.965. Thus visual inspecion of hese funcions can be a firs indicaor of nonsaionariy. Are he auocorrelaions saisically differen from zero? In large samples, if he auocorrelaion is zero, hen he esimaed auocorrelaions ρ ˆ s are approximaely normally disribued wih mean 0 and variance 1 T. For our sample, of size T = 1001, he approximae sandard error is 1 T = 0.0316. A 95% confidence inerval is ±1.96(0.0316) = ±0.062. If a value of ρ ˆ s falls ouside he inerval ( 0.062, 0.062), we conclude ha i is significanly differen from zero. Given our large sample, and correspondingly narrow confidence inerval, he auocorrelaions in Tables 16.2 and 16.3 are saisically differen from zero. When he auocorrelaions are compued hey are cusomarily accompanied by one or more es saisics for he null hypohesis ha all he auocorrelaions ρ s, up o some lag m, are zero. Two commonly repored saisics are he Box-Pierce saisic Slide 16.13
Q m 2 ˆ s (16.3.4) s= 1 = T ρ And a variaion of Equaion (16.3.4) developed by Ljung and Box, ( 2) Q = T T + m 2 ρˆ s (16.3.5) T s s= 1 Under he null hypohesis ha all auocorrelaions up o lag m are zero, he saisics Q 2 and Q are disribued in large samples as χ ( m) random variables. If he value of eiher es saisic is greaer han he criical value from he appropriae chi-square disribuion, hen we rejec he null hypohesis ha all he auocorrelaions are zero and accep he alernaive ha one or more of hem are no zero. Slide 16.14
In Tables 16.2 and 16.3 he column labeled Q-Sa is he Ljung-Box saisic Q. The repored p-values indicae ha for boh series we can rejec he null hypohesis ha all he auocorrelaions are zero. Tesing for zero auocorrelaions is, of course, no acually a es for saionary. The series s2 is a saionary series, wih saisically significan auocorrelaions, as shown in Table 16.2. If we fail o rejec he null hypohesis of zero auocorrelaions, hen we conclude ha he series is a purely random, or whie noise, process, which is a special kind of saionary process. Slide 16.15
16.4 Uni Roo Tess for Saionariy The saionariy of a ime series can be esed direcly wih a uni roo es. The AR(1) model for he ime series variable y is y = ρ y + v (16.4.1) 1 Assume ha v is a random disurbance wih zero mean and consan variance 2 σ v. In he model of Equaion (16.4.1), if ρ = 1 hen y is he nonsaionary random walk, y = y + v, and is said o have a uni roo, because he coefficien ρ = 1. 1 By compuing is variance, we can show ha he random walk process y = y 1 + v is nonsaionary. Suppose ha y 0 = 0, hen, by repeaed subsiuion, Slide 16.16
y 1 1 2 1 2 1 2 3 2 3 1 2 3 = v y = y + v = v + v y = y + v = v + v + v M y = v j= 1 j (16.4.2) Therefore, 2 ( y ) var = σ v (16.4.3) Since he variance of y changes over ime, i is a nonsaionary series. In fac, as he variance of y becomes infiniely large. Recall ha if ρ< 1, hen he AR(1) process is saionary. We can es for nonsaionariy by esing he null hypohesis ha ρ = 1 agains he alernaive ha Slide 16.17
ρ < 1, or simply ρ < 1. The es is pu ino a convenien form by subracing y 1 from boh sides of Equaion (6.4.1), o obain y y =ρy y + v 1 1 1 ( 1) y = ρ y + v 1 =γ y + v 1 (16.4.4) where y = y y 1 and γ=ρ 1. Then H H : ρ = 1 H : γ = 0 0 0 : ρ < 1 H : γ< 0 1 1 (16.4.5) The variable y = y y 1 is called he firs difference of he series y. If y follows a random walk, hen γ = 0 and Slide 16.18
y = y y = v (16.4.6) 1 An ineresing feaure of he series y = y y 1 is ha i is saionary if, as we have assumed, he random error v is purely random. Series like y, which can be made saionary by aking he firs difference, are said o be inegraed of order 1, and denoed I(1). Saionary series are said o be inegraed of order zero, I(0). In general, if a series mus be differenced d imes o be made saionary i is inegraed of order d, or I(d). 16.4.1 The Dickey-Fuller Tess To es he hypohesis in Equaion (16.4.5) we esimae Equaion (16.4.4) by leas squares as usual, and examine he -saisic for he hypohesis ha γ = 0 as usual. Slide 16.19
Unforunaely his -saisic no longer has a -disribuion, since, if he null hypohesis is rue, y follows a random walk. Consequenly his saisic, which is ofen called he τ (au) saisic, mus be compared o specially consruced criical values. Originally hese criical values were abulaed by saisicians Dicky and Fuller. The es using hese criical values has become known as he Dickey-Fuller es. In addiion o esing if a series is a random walk, Dickey and Fuller also developed criical values for he presence of a uni roo (a random walk process) in he presence of a drif. y =α +γ y + v (16.4.7) 0 1 Such series display a definie rend, as we have illusraed wih simulaed daa in Figure 16.1 (e)-(f). This is an exremely imporan case, because as you can see in Figure 16.2, macroeconomic variables ofen exhibi a srong rend. Slide 16.20
I is also possible o allow explicily for a nonsochasic rend. To do so, he model is furher modified o include a ime rend, or ime, y =α +α +γ y + v (16.4.8) 0 1 1 Criical values for he au (τ) saisic, which are valid in large samples for a one-ailed es, are given in Table 16.4. Table 16.4 Criical Values for he Dickey-Fuller Tes Model 1% 5% 10% y =γ y 1 + v 2.56 1.94 1.62 y =α 0 +γ y 1+ v 3.43 2.86 2.57 y =α 0 +α 1+γ y 1+ v 3.96 3.41 3.13 Sandard criical values 2.33 1.65 1.28 Slide 16.21
Comparing hese values o he sandard values in he las row, you see ha he τ- saisic mus ake larger (negaive) values han usual in order for he null hypohesis γ = 0, a uni roo-nonsaionary process, o be rejeced in favor of he alernaive ha γ < 0, a saionary process. To conrol for he possibiliy ha he error erm in one of he equaions, for example Equaion (16.4.7), is auocorrelaed, addiional erms are included. The modified model is 0 1 i i i= 1 m y =α +γ y + a y + v (16.4.9) where ( ) ( ) y = y y, y = y y, K 1 1 2 2 2 3 Slide 16.22
Tesing he null hypohesis ha γ = 0 in he conex of his model is called he augmened Dickey-Fuller es. The es criical values are he same as for he Dickey-Fuller es, as shown in Table 16.4. 16.4.2 The Dickey-Fuller Tess: An Example As an example, consider real personal consumpion expendiures (y ) as ploed in Figure 16.2 (d). This variable is srongly rended, and we suspec ha i is nonsaionary. Inspecion of he correlogram shows very slowly declining auocorrelaions, a firs indicaor of nonsaionariy. We esimae Equaions (16.4.7) and (16.4.8) wih and wihou addiional erms o conrol for auocorrelaion. These resuls are repored in Equaions (16.4.10a). (16.4.10b), and (16.4.10c). Slide 16.23
PCE ˆ = 1.5144 +.0030PCE ( au) (-0.349) (2.557) 1 (16.4.10a) PCE ˆ = 2.0239 + 0.0152 + 0.0013PCE ( au) (0.1068) (0.1917) (0.1377) 1 (16.4.10b) PCE ˆ = 2.111 + 0.00397 PCE 0.2503 PCE 0.0412 PCE 1 1 2 ( au) ( 0.4951) (3.3068) ( 4.6594) ( 0.7679) (16.4.10c) In each case he esimaed value of γ (he coefficien of PCE 1) is posiive, as are he associaed au saisics. Clearly, we do no rejec he null hypohesis ha personal consumpion expendiures have a uni roo. The quesion hen becomes, is he firs difference ( PCE = PCE PCE 1) of he personal consumpion series saionary? Slide 16.24
In Figure 16.4 we plo he firs differences, which cerainly look like he plos of saionary processes in Figure 16.1 (a)-(b). The correlogram shows small correlaions a all lags, suggesing saionariy. 100 50 0-50 -100 70 75 80 85 90 95 DPCE Figure 16.4 Firs Differences of PCE series Slide 16.25
The resul of he Dickey-Fuller es for a random walk (since here is no rend) applied o he series PCE, which we denoe as D, is given in Equaion (16.4.11): Dˆ = 0.9969 D 1 ( au) ( 18.668) (16.4.11) Based on he large negaive value of he au saisic we rejec he null hypohesis ha PCE has a uni roo and accep he alernaive ha i is saionary. Collecing he resuls from he uni roo ess on series PCE is I(1). PCE and PCE, we can say ha he Had he null hypohesis of a uni roo no been rejeced in Equaion (16.4.11), we would have concluded ha PCE is I(2) or inegraed of an order higher han 2. Slide 16.26
16.5 Coinegraion As a general rule nonsaionary ime series variables should no be used in regression models, in order o avoid he problem of spurious regression. There is an excepion o his rule. If y and x are nonsaionary I(1) variables, hen we would expec ha heir difference, or any linear combinaion of hem, such as e = y β β x, o be I(1) as well. 1 2 However here are imporan cases when e = y β1 β 2x is a saionary I(0) process. In his case y and x are said o be coinegraed. Coinegraion implies ha y and x share similar sochasic rends, and in fac since heir difference e is saionary, hey never diverge oo far from each oher. The coinegraed variables y and x exhibi a long erm equilibrium relaionship defined by y =β 1+β 2x, and e is he equilibrium error, which represens shor erm deviaions from he long-erm relaionship. Slide 16.27
We can es wheher y and x are coinegraed by esing wheher he errors e = y β β x are saionary. 1 2 Since we can no observe e we insead es he saionariy of he leas squares residuals, e = y b1 b2x using a Dickey-Fuller es. We esimae he regression ˆ e? 0 e 1 v =α +γ + (16.5.1) where e? = e e 1, and examine he (or au) saisic for he esimaed slope. Because we are basing his es upon esimaed values he criical values are somewha differen han hose in Table 16.4. Table 16.5 Criical Values for he Coinegraion Tes Model 1% 5% 10% e? 0 e 1 v =α +γ + 3.90 3.34 3.04 Slide 16.28
16.5.1 An Example of a Coinegraion Tes To illusrae, le us es wheher y = PCE and x = PDI, where PDI is real personal disposable income (monhly), as ploed in Figure 16.2 (a), are coinegraed. You may confirm ha PDI is nonsaionary. The esimaed leas squares regression beween hese variables is PCE ˆ = 390.7848+1.0160DPI (-sas) (-24.50) (252.97) (16.5.2) Esimaing he Equaion (16.5.1) we obain e? = 0.188250 0.120344 (au) (0.1107) ( 4.5642) e 1 (16.5.3) Slide 16.29
The au saisic is less han he criical value 3.90 for he 1% level of significance, hus we rejec he null hypohesis ha he leas squares residuals are nonsaionary, and conclude ha hey are saionary. We conclude ha personal consumpion expendiures and personal disposable income are coinegraed, indicaing ha here is a long run, equilibrium relaionship beween hese variables. Slide 16.30
16.6 Summarizing Esimaion Sraegies When Using Time Series Daa Le us summarize wha we have discovered so far in his chaper. A regression beween wo nonsaionary variables can produce spurious resuls. Nonsaionariy of variables can be assessed using he auocorrelaion funcion, and hrough uni roo ess. Spurious regressions exhibi a low value of he Durbin-Wason saisic and a high R 2. If wo nonsaionary variables are coinegraed, heir long-run relaionship can be esimaed via a leas squares regression. Coinegraion can be assessed via a uni roo es on he residuals of he regression. There are sill some unanswered quesions. Slide 16.31
1. Firs, if he variables are nonsaionary, and no coinegraed, is here any relaionship ha can be esimaed? In hese circumsances one can invesigae wheher here is a relaionship beween he variables afer hey have been differenced o achieve saionariy. For example, suppose ha he wo variables y and x are I(1) variables, and ha hey are no coinegraed. Since he changes y and x are saionary, we can run regressions of he form y =β +β x + e (16.6.1) 1 2 Esimaing equaions like his one gives informaion on any relaionship beween he changes in he variables. 2. A second case is he one in which y and x are saionary, he implici assumpion mainained for mos of he ex. In his case leas squares or generalized leas squares, whichever is more appropriae, can be used o esimae a relaionship beween y and x. Slide 16.32
3. Finally, here is a hird relaionship ha is of ineres, called an error correcion model, ha can be esimaed when y and x are nonsaionary, bu coinegraed. For I(1) variables, he error correcion model relaes changes in a variable, say y, o deparures from he long-run equilibrium in he previous period ( y β β 1 2x ). I can be wrien as 1 1 y =α +α ( y β β x ) + v (16.6.2) 1 2 1 1 2 1 The changes or correcions y depend on he deparure of he sysem from is long-run equilibrium in he previous period. The shock v leads o a shor-erm deparure from he coinegraing equilibrium pah; hen, here is a endency o correc back owards he equilibrium. The coefficien α 2 governs he speed of adjusmen back owards he long-run equilibrium. We usually expec he sign of α 2 o be negaive, so ha a Slide 16.33
posiive (negaive) deparure from equilibrium in he previous period will be correced by a negaive (posiive) amoun in he curren period. One way o esimae he error correcion model is o use leas squares o esimae he coinegraing relaionship y = β+β 1 2x, and o hen use he lagged residuals e = y b b x as he righ-hand side variable in he error correcion model, ˆ 1 1 1 2 1 esimaing i wih a second leas squares regression. Slide 16.34
Exercise 16.1 16.2 16.3 16.4 Slide 16.35