Econometrics

1. Econometrics Chapter Ten Time-Series Econometric Models

2. Introduction Is it true regression or spurious regression? The practice of classical regression analysis is: first using OLS to estimate the regression model parameter;then based upon the coefficient of determination or the F test statistic to determine the dependency degree between variables and judge the significance of coefficient according to the t statistic of regression coefficient estimates;finally,interpret the economic meaning of regression coefficient estimates based on significantly non-zero regression coefficients.

3. In order to analysis the relation between personal disposable income (I) and personal consumption expenditures (E),using OLS to do linear regression model between E and I.The results is: t = （-7.481） (119.87) From the regression results，R is rather high and t statistic of regression coefficient of personal disposable income I is quite large.It seems that marginal propensity to consume conforms to economic hypothesis.

4. Judge from experience, setting of this model is good and the results are satisfactory.This measuring results are to be used in economic structural analysis and economic forecast. However, it was suggested that, this result may be false! It may only be spurious regression! "To be very careful ! " Is it true regression or spurious regression ?Models, samples, data and test results are satisfactory .why the results are spurious regression?

5. Time-series data is widely used in econometric studies.There are some presuppositions in classical time series analysis and regression analysis, such as stationary and normality of the sequence,etc.When the time-series data of economic variable is directly used in model analysis, the above-mentioned assumptions are included actually. Only when the assumptions are tue,t-test and F-test have relatively high reliability. More and more empirical evidence suggests that most of the time-series data involved in economic analysis is non-stationary.

6. Question:Ifnon-stationary time series are directly used in model analysis as stationary time series,what adverse consequences will be caused? How to determine whether time series is stationary? How to deal with it when non-stationary time series are involved in econometric analysis?

7. This chapter includes: Basic concepts in time-series econometric analysis Unit root test of time series Co-integration

8. 10.1 The Basic Concepts of Time Series 10.1.1 Spurious Regression Problem Assumptions of Traditional Econometric Model: Stationary And Normality of The Sequence The so-called "pseudous regression " means that there is no dependency relationship between variables,but the regression results show dependency relationship exists. In the 1970s,Grange and Newbold found that the fundamental reason of pseudous regression is nonstationary of time series.

9. 10.1.2 Random Process Some random phenomena must be known by studying its process of development and changing.Random process is the dynamic process of random phenomena. For example，to observe and study the number of phone calls each day for a period of time we have to study random variable ξt which depends on the time t, ｛ξt｝is a random process. Another example,GNP of a country in a particular year is a random variable,but when it changes with time,｛GNPt｝is a random process.

10. Defination： If for each specific t(t∈T)，Yt is a random variable.This family random variables {Yt} is called a random process. If T is an interval, then (Yt) is a continuous random process. If T is a discrete set, such as T＝(0，1，2，……)or T＝(……，-2，-1，0，1，2，……), then{Yt} is discrete random process. the random process of discrete time index set is commonly called stochastic time series, time series for short.

11. 10.1.3 the Stationarity of Time Series the so-called stationarity of time series means that statistical law of time series does not change as time goes on. Intuitively,a stationary time series can be seen as a curved line fluctuating around its mean. Theoretically,there are two kinds of stationarity,strictly stationarity and weakly stationarity.

12. Strictly stationary: The joint distribution function of random process｛Yt｝have no reference to time displacement.Set｛Yt｝is a random process , n, h are arbitrary real numbers, If the joint distribution function to meets: Then {Yt} is a strictly stationary random process, It’s distribution structure does not change over time.

13. Weakly stationary: Expectation, variance and covariance of random process {Yt} do not change over time.If｛Yt｝meets: E（Yt）=μ， Then｛Yt｝is weakly stationary random process. In the general analysis , stationarity usually refers to a weak stationary random process.

14. Non-stationary time series : Statistical law of time series changes over time. Characteristics of the random process generating multivariate time series data change over time. Time-series data encountered in actual are likely to be non-stationary series,however, stationarity has an important position in econometric modeling.So stationarity test for the observed values of time series data is necessary.

15. 10.2 the Unit Root Test 10.2.1 Unit root process In order to illustrate the concept of unit root process, We have focused in AR (1) model: Yt=φYt-1+εt to analysis. According to the theory of stationary time series analysis we can see,when ,{Yt} is stationary, This model is the classic Box-Jenkins time series AR (1) model.

16. When ，the sequence generation process changes into the following random walk process : Yt=Yt-1+εt of which is Iid,zero mean and variance constant for the .Variance of random walk process is as follows: when ,variance of the sequence tends to infinity,note random walk process is non-stationary.

17. If a sequence is random walk process, this sequence is called a "unit root process ". Why is it called "unit root process"? A first-order autoregressive model is expressed as the following forms: or Of which L is the lag operator. That is,

18. According to the lag polynomial of the model , the corresponding linear equation is: (Often referred to as the characteristic equation) The root of the equation is : when sequence is stationary.The root of the characteristic equation satisfy the condition ; when , sequence generation process changes into a random walk process. The root of the characteristic equation z=1. So series are usually called to contain unit root or sequence generation process is called for the "unit root process."

19. By the former we can see that random walk process is non-stationary. • Thus,it turns to test whether the characteristic equation has a unit root. This is the origin of unit root test methods. From the definition of the unit root process we can see that first order difference of a process with a unit root: is a stationary process.Process like this which is changed into stationary process after first difference transformation is called integrated of oder one.It is noted as ~I (1).

20. Sometimes, a sequence is still nonstationary after first difference transformation.If the sequence becomes stationary after second order difference transformation, it is called integrated of oder two and noted as ~I(2). In general, if the sequence becomes stationary through d times difference transformation while is nonstationary after d-1times difference,it is called integrated of order d,noted as ~I(d) ,d is integer order. In particular, If the sequence itself is stationary,it is called integrated of oder zero,noted as ~I (0).

21. Two. Dickey-Fuller Test（DF Test） Most of the economic variables show a strong trend characteristics. These economic variables with trend characteristics usually occur two kinds of cases after economic shocks or economic impact.One is that economic variables gradually go back to their long-term trend track.The other is that they are in the state of random walk instead of going back to their long-term trend track. If the economic variables we study follow a non-stationary process, the regression may lead to spurious regression results. This is the significance of unit root test study.

22. Suppose the data sequence is generated by the following autoregressive model : Among them , are identically distributed which are independt from each other and the expection value is zero ,the variance is at this time ,we should examine the sequence whether it contains a unit root. We suppose The OLS regression coefficient is Test statistics is

23. Under the condition of ,the t statistic as follows : • But the trouble is, Dickey Fuller found that in the case of the former assumption, the statistic do not obey t distribution. As the t test statistic is no longer subject to the traditional t-distribution, so the traditional t-test failure. • It can be shown that if the limit distribution of the above-mentioned statistics exists, we generally called it Dickey-Fuller distribution. In order to distinguish it from the the DF test, t statistic values can sometimes also called value .

24. Dickey Fuller use the critical value of DF tests to compile a DF test critical value table for inspection.We can compare statistical t value threshold with the DF test to reject or accept the original hypothesis under certain level of significance . • The steps are as follows: • 1. With the observational data, we use the OLS estimation method to establish first-order autoregressive model, • and obtain the OLS estimatiors of regression coefficients.

25. 2. We suppose : • the T statistic is • 3. We calculate the t statistics under the original assumption ,then compare the value of t statistics with the DF test critical value : If the t statistics value is less than the critical value of DF tests, then refused the original hypothesis, indicating there is no sequence of unit root; If the t statistics value is greater than or equal to the DF test the critical value, then accept the original hypothesis, indicating the existence sequence unit root;

26. In addition, Dickey Fuller found that , the DF critical value of the test is related to the data generation with the sequence of processes as well as the type of regression model, so they prepared the different critical value of the table for the following three kinds of equations . Later, the Mackinnon expanded the table, forming a currently using a wide range of critical values of the table which is also used in the Eviews. • The three models are : • (1) (2) • (3)

27. 10.3 The Augmented Dickey-Fuller Test • The problem of DF test is that we assume there is no random disturbance autocorrelation in the model, . However, most economic data series can not satisfy this assumption, when there is random disturbance autocorrelation, the direct use of DF-test bias will appear, in order to ensure the effectiveness of the unit root test, people form the extended DF tests (Augmented Dickey-Fuller Test), called the ADF test. • The three model of DF test are : • (1) (2) • (3) • In which is a random disturbance term, it can be a general stationary process .

28. In order to use DF test method, the model becomes the following formula: • (1) • (2) • (3) • It can be shown that the limit distribution of the t statistics under the original hypothesis is equal to the limits of the distribution of DF testing, and thus we can use the same threshold table, this test is known as the ADF test .

29. Example 10.1 : According to the China statistical yearbook of China in 2004, we can get the GDP of 1978-2003, then we can check whether it is a stationary sequence.With the help of EVIEWS, we can get the results in table data 10.1, and the timing diagram in figure 10.1. • table 10.1 the GDP of China in 1978~2003

30. Graph 10.1 the Time Series of GDP

31. From the graph of GDP, we can find out there exist trend in the sequence, so choose the third model to do the ADF test. • Estimation results are as follows: • Under the original assumption ,the unit root value of t statistics is

32. In the three significant level of 1%, 5%, 10%, the Mackinnon critical value of unit root test are -4.4167, -3.6219 and -3.2474.Apparently, the value of t statistics is greater than the corresponding, and thus we can not refuse the orginal assumption , so we say the series of GDP in 1978-2003 has a unit root, and it is a nonstationary sequence.

33. 10.3 The Cointegration • 10.3.1 the Concept of Cointegration • There is an example of the demand for money. The classic theoretical analysis tells us that the demand of money in a country or a region depends largely on the size of variables and the opportunity cost variables, namely, real income, price level and interest rates. If we use the Logarithmic from to establish the econometric model to describe it ,we can get the following model : • Provide that ,M=the demands of money,p= the price level,Y=the real income ,r =the interest rate,u= the disturbance

34. Usually,people are concerned about whether the estimated money demand function reveals the long-run equilibrium relationship between the demand for money. • If the money demand function is appropriate, then the deviation among the demand for money on the long-run equilibrium relationship will be temporary, the disturbance sequence is a smooth sequence, the money demand function reveals the long-run equilibrium relationship between the demand for money. On the contrary, if the disturbance sequence has stochastic trend and showed the phenomenon of non-stationary, then the model error will gradually accumulate, making deviation will not disappear in a long period. Therefore, whether the money demand model is of practical value, the key lies in the disturbance sequence is smooth or not.

35. Money supply, real income, price levels and interest rates may be I (1) sequence. Under normal circumstances, a number of non-stationary series is a linear combination of non-stationary series. If the money supply, real income, price levels and interest rates, any linear combination are non-stationary, then the money demand model of the disturbance sequence can not be stable, and thus model and did not reveal the relationship between long-term stability of money demand . Conversely, if the money demand model describes the long-run equilibrium relationship between the demand for money, then the sequence of the disturbance must be a smooth sequence, that is, non-stationary money supply, real income, price levels and interest rates between the four variables smooth linear combination.

36. The above examples reveal to us the fact that: " a balanced system which contains non-stationary variables necessarily mean that the combination of some non-stationary is smooth." • This is the ideas of cointegration theory. • The cointegration refers to a certain number of non-stationary linear combinations of variables is smooth. For example, the income and consumption, wages and prices, government spending and taxes, exports and imports and so on, these economic time series generally are non-stationary series, but they often exist the long-run equilibrium relationship between them.

37. Here are a strict definition of cointegration: • As to the two series and ,if • and there are two non-zero constants ,makes that • then ,we say there is cointegration between . • Generally, set there are k number of series • use to describe the k uyghur vector series composed by that k number of series. If, (1) every series in is the d orders integration series, that is ; • (2)exist the nonzero vector ，making

38. is the (d-b) orders integration series, that is . And we call the component of the vetor series is the d,b orders integration, recorded as , and the vetor is called as the integration vector.

39. Specially, if d=b=1, ，that is to say, since the component series are non-stationary first-order integration series, but there will be some linear combination is stationary. This kind of（1，1）-order cointegration is common in the econometric analysis. For example, supposing there is (1,1)-order cointegration relationship between variable and variable , cointegration variable is , so this kind of cointegration can be showed as, (10.10) And the combination variable is the I（0）process.

40. The conception of cointegration is rather important to establish the econometric model using non-stationary variables, and test the long-run equilibrium relation between these variables. （1）If there is the cointegration relationship between many non-stationary variables, so these variables can be composed into a stationary series. And this stationary series can be used to describe the equilibrium relation btween the primary variables.

41. （2）If and only if there is the cointegration between the many non-stationary variables, the regression model is meaningful established by these variables. So the cointegration test is the effictive method to distinct the real regression and the spurious regression. • （3）The non-stationary variables which have the cointegration relationship can be used to estalishe the error correction model. Because the error correction model combines the long-term relationship and short-term dynamic characters into one model, it not only can solve the problen that we always ignore the spurious regression in the tradional econometric model, but it also can overcome the weakness that we ignore the level variable in difference model.

42. 10.3.2, the Test of Cointegration There are two ways to test the cointegration, ●based on regression residual cointegration test, this test is also called the cointegration test for single equation. ●based on regression coefficient cointegration test including complete imformation, Here, we only consider the situation of single equation, and introduce the EG two-step test for the cointegration relationship between two variables.

43. Ⅱ Cointegration Test There are two methods to test cointegration: ● The Cointegration Test based on the residual of regression, which is known as a single equation cointegration test ●Complete Information Cointegration Test based on the regression coefficient. Here we only consider the case of a single equation, and focuses on the E-G two-stage test method of two variables cointegration.

44. E-G two-step test： Step One：If and are all first－order integration series, ～ （I（1））, ～ （I（1）），that means that and are stationary series. Using OLS method estimates regret ion equation： estimating and getting residual series：

45. Step two，test the stationary of . If is stationary，Xt and Yt are co-integration，otherwise are not co-integration。Because if Xt and Yt are not co-integration , their any linear combination is not stationary, that the residual series is not stationary。In another word, the test of stationary of the residual is mean to the test of the co-integration of Xt and Yt .

46. There are two methods that test the residual series are not stationary: The first method is the DF test to examine the residual, that is also named the unit root test, and the method has been introduced in the previous . But to be noted that, the threshold of DF test and ADF test must be the special critical value table that be established by Engle-Granger .

47. The other method is the DW test based co-integration regression. Detailingly, using the co-integration regression residual makes the DW statistics: If is Random Walk, that the mathematical expectation of is zero, so the value of DW is approaching zero. Therefore, to test , if the assumption is right, so is Random Walk , that also means that Xt and Yt are not co-integration, otherwise are co-integration.

48. Sargan and Bhargava established the critical value table used in co-integration test earliest. Table 10.2 is the critical value table about 100 observations. For example, if DW value is 0.71, under the significance level of 1% ,we can refuse, that means to refuse the no co-integration hypothesis .

49. Ⅲ Error Correction Model (ECM) Error Correction Model is usually called ECM for short, and is a kind of specific forms of econometric models. There are usually two steps to establish ECM, we will respectively establish econometric models based on the long-term characteristics and short-term characteristics of dates.

50. Step one: to establish the long-term relationship models. That to estimate the relationship between time series variables through the level variables and the OLS method . If the estimation results format a stable residual series, then there exists co-integration relationship between these variables, and the selection of variables of the long-term relationships model is reasonable, and the regression coefficients makes economic sense.