Cointegration Approach to the Spurious Regression Model 1 Olatayo T.O, 2 Adeogun A.W and Lawal G.O

Cointegration approach to the spurious regression model 1 Olatayo T.O, 2 Adeogun A.W and Lawal G.O. 1Department of Mathematical Sciences, Olabisi Onabanjo University, Ago-Iwoye, Ogun State Nigeria. 2Department of Mathematics and Statistics, Yaba College of Technology Lagos State, Nigeria

ABSTRACT

In this paper we investigated the concept of co-integration methods as an approach to spurious regression model. The effect of employing arbitrary differencing method to detect which spurious regression will result from a true model with time series economic data was carried out. It was found that the concept of co-integration test is a more articulate procedure of determining variables whose spurious will result from truly related variables.

Keywords; cointegration, spurious regression, differencing method and time series economic,data.

INTRODUCTION problem. Dynamic ordinary least squares (OLS) and general linear (GLS) estimators were proposed by When the Stochastic trends of two or more difference Stock and Watson (1993), Phillips and Lovetan stationary variables are eliminated by forming a linear (1991) and Shaikkonen (1991). Park and Ogaki; combination of these variables, the variables are said (1991) developed a method of seemingly unrelated to be cointegrated in the technology of Engle and canonical cointegrating regressions (SUCCR) for this Granger (1987). case and Engle and Granger’s (1987) augmented When there is one cointegrating vector, a regression Dickey-filler (ADF) test applied the said Dickey test to of one variable in Xt on the others is called a the residual from cointegrating regressions. cointegrating regression, and when there is no Co- Shangodoyin, Adeogun and Olatayo (2008) also integrating vector, a regression of one variable in X t considered the effect of non-robustness in the on the others is called a spurious regression. spurious regression model and concluded that the Spurious regressions have a long history in statistics, violation of the assumptions play an important role in dating back at least to Yule (1926). Therefore, determining if a spurious regression emanates from spurious relationship referring to a correlation the statistically related model for a reliable predictive induced between two variables that are casually purposes. Niels Haldrup and Michael Jasson (1999 & related but both dependent on other common 2000) introduce a unifying approach to spurious variables. This is accomplished either by improving regression, Co-integration and Near cointegration. the function of time regressions or by subtracting a They concluded that the notion of near co-integrating function of time from all series used. helps to bridge the gab between the polar cases of spurious regressions and co-integration. Theoretically, spurious regression occurs when the variables being regressed are integrated variables of Nevertheless, the above authors and many others order one i.e. I(1), in which case they are not have not taken the concept of cointegrating approach stationary, but stationary if difference once, Wen-Jen to spurious regression models into consideration. Tsay (1999). Almost all economic variables one I(1), The purpose of this paper is to consider the concept hence expected to lead to spurious regression. Some of cointegration approach to spurious regression of the examples of the spurious regression results is model with time series economic data. The method in Darlauf and Phillips (1988), Granger and Newbold were analysed empirically with exchange rate data of (1974), Ogaki and Choi (2001). The asymptotic U.S dollar to Nigeria Name and U.S dollar to Great theories of Philips (1986, 1987, 1998) have been Britain pound sterling. The exchange range data of used to understand the spurious regression problem, financial currency is a suitable financial economic but have not been used to provide a solution to the indicator and measure of the performance of the Am. J. Sci. Ind. Res., 2012, 3(5): 263-269

Nation’s economy. The method of differencing was Fuller(ADF) and Phillips-Perron test procedures, arbitrarily employed to detect which spurious Engle and Granger (1987) are most prominent regression will result from a time model or otherwise, empirical test that are used in modeling a series with without considering if the series are in the first place identified unit root(s). generated from an I(1) process. The concept of Unit root testing procedures: In general, the cointegration testing is a more articulate procedure of procedures test a data generating process for determining variables whose spurious regression will difference stationary (trend non-stationary) against result from truly related variables. trend stationary. The requirements for the test are: MATERIALS AND METHODS (i). The models which are:

If Xt and Yt are non-stationary, but I(1) variables, and Yt =  + t + (-1)Yt-1 + t------eqn(2) that a linear combination of Xt and Yt as specified If augmented the model becomes below will result in a stationary process, then Xt and Model I

Yt are said to co-integrates. J Yt = (-1)Yt-1 + jYt-j + t (no intercept Xt =  + Yt + t------eqn(1) j=1 where t is a white noise process or linear trend) Model II In cointegration analysis, stationary of a series is very basic and this is ascertained by implementing the unit J Yt =  + (-1)Yt-1 + jYt-j + t (no linear trend) root testing procedure. j=1 Stationary series and unit root: A time series is Model III stationary, if its mean, variance and autocovariances J are independent of time. For a stationary time-series Yt =  + t + (-1)Yt-1 + jYt-j + t------eqn(3) Xt the following properties must be satisfied: j=1

(i). E(Xt) =  (ii). Sample size (n) and 2 (iii). The level of significance (e) The test (ii). E[(Xt) ] = Var(Xt) = (0) procedure admit the following: Augmented Dickey-Fuller t-Test Procedure (iii). E[(Xt)(Xt-k)] = Cov(Xt,Xt-k) = (k) , k = 1, 2, … Model I J Cov(Xt,Xt-k) (k) Yt = (-1)Yt-1 + jYt-j + t------eqn(4) (iv). Corr(Xt,(Xt-k) = k = = j=1 Var(Xt) (0)

, k = 1, 2, … Hypothesis: H0 :  = 1 A time series of random error (a measure of H1 :  < 1 disequilibrium) with the characteristics of errors being p-1 Test Statistics: t = small on the average and stabilises over time is a se(p) near example of a stationary series. A non-stationary where p is the estimated  series will essentially exhibit trend in its mean over Critical Value: ADFt(I,N,e) time and if such trend is stochastic, it implies that a Model II first difference operation on the series will make it J trendless and hence stationary satisfying the Yt =  + (-1)Yt-1 + jYt-j + t------eqn(5) properties stated above. A series with this behaviour j=1 is said to possess a single “unit root”, and where the Hypothesis: H0 :  = 1 H0 :  = 0, given  = differencing required before stationary is attained is 1 more than once, then the series will have a multiple H1 :  < 1 H1 :   0 unit roots. p-1 a Test Statistics: t = t = Identifying a series with one or more unit root is se(p) se(a) purely empirical although economic theories sometimes give prior information that suggests its where p is the where a is the existence. The Dickey-Fuller, Augmented Dickey- estimated  estimated  Critical Value: ADFt(II,N,e) ADFt(II,N,e)

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Model III If  < 1 then stop (no unit root) else conclude J (unit root)! Yt =  + t + (-1)Yt-1 + jYt-j + t------eqn(6) Unit root Test for Exchange Rate of US$ to NGN j=1 (DNt)

Hypothesis: H0 :  = 1 H0 :  = 0, given  = Step 1 Model III 1 H0 :  = 0, given  = 1 Y =  + t + (-1)Y +  Y +  Y +  Y + H1 :  < 1 H1 :   0 H1 :   0 t t-1 1 t-1 2 t-2 3 t-3 p-1 a t------eqn(10) Test Statistics: t = se(p) t = se(a) t = Testing the hypotheses b H0 :  = 1 se(b) H1 :  < 1 at 0.05 level of significant where p is the where a is the where b is Test Statistics the p-1 -0.354 t = = = -0.369 estimated  estimated  estimated  se(p) 0.96 Critical value Critical Value: ADFt(III,N,e) ADFt(III,N,e) ADFt ,III,100,0.05 = -0.90 ADFt(III,N,e)  The steps are enumerated below: Decision Step 1: Reject H0 if t > ADFt,III,100,0.05 1. Estimate Model III H0 is not rejected   = 1 J Yt =  + t + (-1)Yt-1 + jYt-j + t------eqn(7) Testing  = 0 given  = 1 j=1 H0 :  = 0 H : 0 at 0.05 level of significant 2. Test  = 1 using ADFt distribution 1   If  < 1 then stop (no unit root) else Test Statistics continue b -0.000458 t = se(b) = 0.00 3. Test  = 0 given  = 1, using ADF t If   0, then  t is infinitely negative Test  = 1 using the normal distribution: Critical value If  < 1 then stop (no unit root) else conclude (unit ADFt,III,100.0.05 = 2.79 root)! Decision else continue Reject H0 if t > ADFt,III,100,0.05 Step 2: H0 is not rejected 4. Estimate Model II   = 0 Presence of unit root, with no linear trend J Step 2: (Testing for presence of drift) Yt =  + (-1)Yt-1 + jYt-j + t------eqn(8) j=1 Model II 5. Test  = 1 using ADF distribution Yt =  + (-1)Yt-1 + 1Yt-1 + 2Yt-2 + 3Yt-3 + t------t If  < 1 then stop (no unit root) else continue ------eqn(11) Testing the hypotheses 6. Test  = 0 given  = 1, using ADF t H :  = 1 If   0, then 0 H :  < 1 at 0.05 level of significant Test  = 1 using the normal distribution: 1 Test Statistics If  < 1 then stop (no unit root) else conclude p-1 -0.136 (unit root)! t = se(p) = 0.061 = -2.2295 else continue Critical value Step 3: 7. Estimate Model I ADFt,II,100.0.05 = -0.05 Decision J Reject H0 if t > ADFt,II,100,0.05 Yt = (-1)Yt-1 + jYt-j + t------eqn(9) j=1 H0 is not rejected   = 1 5. Test  = 1 using ADF distribution t Testing  = 0 given  = 1

H0 :  = 0

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H1 :   0 at 0.05 level of significant Decision Test Statistics Reject H0 if t > ADFt,III,100,0.05 a 0.648 H is not rejected t = = = 2.2268 0  se(a) 0.291   = 0 Critical value Presence of unit root, with no linear trend

ADFt,II,100,0.05 = 2.54 Step 2: (Testing for presence of drift) Decision Model II

Reject H0 if t > ADFt,II,100,0.05 Zt =  + (-1)Zt-1 + 1Zt-1 + 2Zt-2 + 3Zt-3 + t------H0 is not rejected ------eqn(14)   = 0 hence no drift Testing the hypotheses Step 3: H0 :  = 1 Model I H1 :  < 1 at 0.05 level of significant Yt = (-1)Yt-1 + 1Yt-1 + 2Yt-2 + 3Yt-3 + t-----eqn(12) Test Statistics Testing the hypotheses p-1 -0.108 t = = = -1.9636 H0 :  = 1 se(p) 0.055 H1 :  < 1 at 0.05 level of significant Critical value Test Statistics ADFt,II,100.0.05 = -0.05 p-1 0.000216 Decision t = = = -0.216 se(p) 0.001 Reject H0 if t > ADFt,II,100,0.05 Critical value H0 is not rejected

ADFt,I,100.0.05 = 1.29   = 1 Decision Testing  = 0 given  = 1

Reject H0 if t > ADFt,I,100,0.05 H0 :  = 0 H0 is not rejected H1 :   0 at 0.05 level of significant   = 1 Test Statistics Unit root Test for Exchange Rate of GB£ to NGN a 0.553 t = = = 1.9540 (PNt) se(a) 0.283 Step 1 Critical value

Model III ADFt,II,100,0.05 = 2.54 Zt =  + t + (-1)Zt-1 + 1Zt-1 + 2Zt-2 + 3Zt-3 + Decision

t------eqn(13) Reject H0 if t > ADFt,II,100,0.05 Testing the hypotheses H0 is not rejected H0 :  = 1   = 0 hence no drift H1 :  < 1 at 0.05 level of significant Step 3: Test Statistics Model I p-1 -0.348 Z = (-1)Z +  Z +  Z +  Z +  t = = = -3.59 t t-1 1 t-1 2 t-2 3 t-3 t----- eqn (15)  se(p) 0.097 Testing the hypotheses

Critical value H0 :  = 1 ADFt,III,100.0.05 = -0.90 H1 :  < 1 at 0.05 level of significant Decision Test Statistics Reject H0 if t > ADFt,III,100,0.05 p-1 0.000224 t = = = -0.224 H0 is not rejected se(p) 0.001   = 1 Critical value

Testing  = 0 given  = 1 ADFt,I,100.0.05 = 1.29 H0 :  = 0 Decision H1 :   0 at 0.05 level of significant Reject H0 if t > ADFt,I,100,0.05 Test Statistics H0 is not rejected b -0.000551   = 1 t = se(b) = 0.00

 t is infinitely negative Critical value

ADFt,III,100,0.05 = 2.79

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Unit root Test for Exchange Rate of US$ to GB£ To establish this, the cointegration test is carried out. (DPt) When variables do not cointegrates, their regression Step 1 equation retains the unit root property. There are two Model III major approaches to conducting the cointegration

Xt =  + t + (-1)Xt-1 + 1Xt-1 + 2Xt-2 + 3Xt-3 + test, these are the Engle-Granger approach and the t------eqn(16) Johansen approach. In this paper the Engle-Granger Testing the hypotheses approach will be adopted.

H0 :  = 1 The engle-granger approach cointegration test PROCEDURE H1 :  < 1 at 0.05 level of significant Test Statistics Considering variables Xt and Yt which are p-1 -0.167 I(1), a regression equation of their linear combination t = se(p) = 0.057 = -2.93 is given by: Critical value Yt =  + Xt + t Since Xt and Yt are I(1), if it can be shown that t is ADFt,III,100.0.05 = -0.90 Decision I(0), then the set of variables [Yt,Xt] will cointegrate. A Reject H if t > ADF ,III,100,0.05 unit root test procedure for the residual of the model 0  t thus specified: H0 is not rejected   = 1 Model I Testing  = 0 given  = 1 J t = (-1)t-1 + t-jt-j + ut (An augmented H0 :  = 0 j=1 H1 :   0 at 0.05 level of significant Test Statistics model) b 0.00004895 Hypothesis: H0 :  = 1 t = =  se(b) 0.00 H1 :  < 1 p-1  t is infinitely positive Test Statistics: t = Critical value se(p) where p is the estimated  ADFt,III,100,0.05 = 2.79 Decision Engle-Granger Cointegration Test for the Exchange Rate of US$ to NGN (DNt) and Reject H0 if t > ADFt,III,100,0.05 H is rejected US$ to GB£ (DPt) 0 Model    0 Presence of linear trend lnDNt =  + lnDPt + t------eqn(17) The unit root test for the residuals model is specified Testing  = 1 using the t-statistics from estimating the thus: augmented version with critical values taken from the standard normal table t = (-1)t-1 + t-1t-1 + t-2t-2 + t-3t-3 + ut Testing the hypotheses H0 :  = 1 H0 :  = 1 H1 :  < 1 at 0.05 level of significant Test Statistics H1 :  < 1 at 0.05 level of significant t = -2.911 Test Statistics Critical value p-1 -0.219 t = se(p) = 0.072 = -3.041 z0.05 = 1.65 Decision Critical value To use the J. G. MacKinnon’s table of critical values Reject H0 if t > z0.05 of cointegration test, the regression model is with no H0 is not rejected   = 1 and   0 trend, hence Model 2 is adopted and use the expression: Cointegration test: Considering a set of M variables 2 1 1 Xt (a 1M vector). If Xt ~ I(1), the column-wise linear     CV(K,Model,N,Sig) = b + b1 N + b2 N combination of Xt is again usually I(1). The set of     variables Xt is said to cointegrates if a linear From J. B. MacKinnon’s table, where K = 1, Model = combination of the variables will result in a stationary 2, N = 99 process i.e. I(0). A regression relation involving Xt and Significant level = 0.05 variables will only be meaningful, if they cointegrates. b = -2.8621, b1 = -2.738 and b2 = -8.36

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 1  i.i.d. The test adopts a step by step procedure in  CV(1,2,99,0.05) = -2.8621 – 2.73899 + testing for presence or otherwise of a unit root. The 2 variable is generated by an I(1) process and  1  b299 = -2.8621 – 0.0276 possesses a linear trend with the possibility of a drift. = -2.890 From the unit root tests, all the variables are Decision confirmed to be generated by I(1) process, implying Reject H0 if t > CV(1,2,99,0.05) that there regression is bound to be spurious. A linear H0 is not rejected combination of any of the variable can thus be tested  = 1 i.e. unit root is present for models that will lead to meaningful spurious Engle-Granger Cointegration Test for the regression or otherwise. The exchange rates of US Exchange Rate of US$ to NGN (DNt) and Dollar to NG Naira and GB Sterling pound to NG GB£ to NGN (PNt) Naira does not cointegrates, thus their linear Model regression model will be meaningless. lnDN =  + lnPN +  t t t------eqn(18) The result of the cointegration test showed that the The unit root test for the residuals model is specified variables DN /DP and DN /PN does not cointegrates, thus: t t t t so regressing their linear equations will not give a t = (-1)t-1 + t-1t-1 + t-2t-2 + t-3t-3 + ut satisfactory estimation economically, since it will be Testing the hypotheses non-stationary. H0 :  = 1 H1 :  < 1 at 0.05 level of significant The reason for this may not be far fetched, initial Test Statistics examination of the nature of the variables shows that, p-1 -0.18 the assumptions under regression analysis does not t = se(p) = 0.059 = -3.0508 hold, strongly suggesting that the variables regressed Critical value will exhibit basic relational problems hence CV(1,2,99,0.05) = -2.890 statistically not adequate. Therefore equation (20) Decision give the Error Correction Model, with which the variables can be regressed like one generated by an Reject H0 if t > CV(1,2,99,0.05) H is not rejected I(0) process, and the result is significant economically 0 and statistically.  = 1 i.e. unit root is present Error Correction Model CONCLUSION Suppose the variables tested cointegrates, then Although the 1st difference method of detecting a Y =  + X +  t t t spurious regression suggests that DN /PN may lead  = (-1) + u t t t t-1 t to true model spurious regression, the cointegration where  < 1 and ut is stationary. analysis indicated otherwise. It can thus be inferred The short-run dynamics of the model will be that the cointegration test is more sensitive to Yt = Xt + t------eqn(19) detecting variables that violated regression analyses = Xt + (-1)t-1 + ut assumptions and that had the assumptions not Y = X + (-1)( Y --X ) + u t t t-1 t-1 t violated, DNt/PNt will most likely be a true model ------eqn (20) whose regression will be spurious.

The study revealed in general that it is not enough to RESULTS AND DISCUSSION focus on whether a regression will be spurious when Essentially Dickey-Fuller test assumes that the error regressed or that a spurious regression will result terms is anindependently and identically distributed from a true or false model. It is equally important to (i.i.d) process. This assumption is relaxed in the ascertain if the variables to be analysed, will result in Phillips-Perron test. The Augmented Dickey Fuller a non-robust model. When regression models and the assumptions made for their adoption are (ADF) test procedure is specified when t is autoregressive to eliminate serial correlation of specified, the fact that whether it will be spurious or errors. The test is applied, considering the limiting not can adequately be examined with necessary test distribution and critical values, it is thus shown that procedures like the cointegration test. the assumption that t ~ i.i.d will remain valid. ADF test procedure embraces a wider consideration of t ~

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