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Testing for Unit Root in Macroeconomic of China

Xiankun Gai Shan Dong Province Statistical Bureau Quan Cheng Road 221 Ji Nan, China [email protected];[email protected]

1. Introduction and procedures

The immense literature and diversity of unit root tests can at times be confusing even to the specialist and presents a truly daunting prospect to the uninitiated. In order to test unit root in macroeconomic time series of China, we have examined the unit root throry with an emphasis on testing principles and recent developments. Unit root tests are important in examining the stationarity of a time series. Stationarity is a matter of concern in three important areas. First, a crucial question in the ARIMA modelling of a single time series is the number of times the series needs to be first differenced before an ARMA model is fit. Each unit root requires a differencing operation. Second, stationarity of regressors is assumed in the derivation of standard inference procedures for regression models. Nonstationary regressors invalidate many standard results and require special treatment. Third, in analysis, an important question is whether the disturbance term in the cointegrating vector has a unit root. Consider a time series data as a data generating process(DGP) incorporated with trend, cycle, and seasonality. By removing these deteriministic patterns, the remaining DGP must be stationry. “Spurious” regression with a high R-square but near-zero Durbin-Watson statistic, often found in time series litreature, are mainly due to the use of nonstationary data series. Given a time series DGP, testing for is a test for stationary. It is also called the unit root. Testing for the problem of unit root for each time series is more of a process than it is a step. If a problem is identified, the original data is differenced and test again. In this way, we are able to identify the order of the integrated process for each data series. Once each data series has completed this process they are regressed together and test for a cointegrating relationship. Since the tests we used, Dickey- Fuller(DF) and Augmented Dickey-Fuller(ADF) require the error structure of model to be individually independent and homogeneously distributed, anomalous serial correlation in time series must be treated before these test can be applied. Therefor, serial correlation is tested and corrected as the pretest step of each process. Instead of directly correcting the error structure, we will modify the dynamics of the data generating process with logged dependent variables. To carry out both the DF and ADF tests for unit root, we test for the most complicated problem first and then simplifying our model if problems are absent. A hierarchy of three models is formulated and tested. First, we estimated the Random Walk Model with trend, and drift, as follows:

(1) ∆X = α + βt + (ρ −1)X − + ∑ ρ X − + ε t t 1 i=1,2,L i t i t

Using augmented lags of dependent varisble ∑ ρ X − is to ensure a white noise ε for the test i=1,2,L i t i t process. Testing for the zero-value coefficient of Xt-1 ,or equivalently =1, is the focus of the unit root tests. If unit root is not found in the first model, we continue the process by estimating the Random Walk Model with drift, as follows:

(2) ∆X = α + (ρ −1)X − + ∑ ρ X − + ε t t 1 i=1,2,L i t i t And finally, the Random Walk Model ,as follows:

1 (3) ∆X = (ρ −1)X − + ∑ ρ X − + ε t t 1 i=1,2,L i t i t Testing for unit root is the first step of time series model building. For a univariate case, several versions of DF and ADF test are available. For multivariate time series, after testing unit roots for each variable, a cointegration test should be carry out to ensure that the multiple regression model is not spurious. Critical value for the DF unit root test are based on t-statistic or F-statistic. Regression will calculate the t-statistic or F-statistic for the null hypothesis : =1. The t-statistic or F-statistic, however, cannot be reffered to the critical values in the standard t table or F table, since under the null hypothesis the left-hand variable is nonstationary. MacKinnon has supplied a more clear DF critical values table.

2.Data and Empirical results

We use SAS/ETS testing for unit root in macroeconomic time series of China. Data and some result as follows(sample period is from 1950 to 1998): X1-indices of gross domestic product(1950=100, The data are calculated at comparable prices), X1 is I(2), LOG(X1) is I(1). X2-indices of gross capital formation, X2 is I(2),LOG(X2) is I(1). X3-indices of final consumption expenditure, X3 is I(2), LOG(X3) is I(1). X4-per capita net income of rural households, X4 is I(2), LOG(X4) is I(1). X5-General retail price index, X5 is I(2), LOG(X5) is I(1). X6-General purchasing price index of farm products. X6 is I(2), LOG(X6) is I(1). X7-General consumer price index. X7 is I(2). LOG(X7) is I(1). X8-deposits. X8 is I(1). X9-loans.X9 is I(1). X10-total production of energy. X10 is I(1). X11-per capita income in urban residents. X11 is I(1). If d of I(d) can take non-integrated ,the degree of unit root would be described.

REFERENCES

Dickey, D. A. and Fuller, W. A. (1979). Distribution of the Estimators for Autoregressive Time Series with a Unit Root. JASA, 74,427-431.

Engle, R. F. and Granger, C. W. J. (1987). Cointegration and Error Correction: representation, estimation and testing. Econometrica, 55,2,251-276.

MacKinnon, J. (1991). Critical Values for Cointegration Tests, in R. F. Engle et al ,Long-run Economic Relationships: Readings of Cointegration. Oxford: Oxford University Press, 267-276.

Pillips, P.C.B.(1986). Understanding Spurious Regressions in . Journal of Econometrics, 33,311-340.

FRENCH RÉSUMÉ

Ce papier fait une discussion de la élément racine(unit root) dans la série de temps macroéconomique dans la Chine.

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