CHAPTER 3 RESEARCH METHOD AND DATA INFORMATION
3.1 M ETHODOLOGY
The purpose of this study is to explore the dynamic relationship between GDP, consumption and other variables. As the assumptions of classical regression model necessitate that all sequences be stationary I(0) and that errors have a zero mean and a finite variance. In the presence of many economic series appear to have a non-stationary component, there might be statistical bias results a spurious regression.7
Therefore, before present study of Cointegration Test or Vector Autoregression, we use unit root test to determine whether the variable is stationary or not. If all the series are stationary, we employ VAR to analysis. If all the series are non-stationary, we employ the Cointegration Test to analysis the variables of a long-term stable equilibrium relationship, and if they have the cointegration relationship, Vector Error Correction Model (VECM) is applied for further study.
Finally, we use Impulse Response Function, given known conditions, an unexpected change was made to compare with the original unchanged environment. Through the observation on differences between two expected values, the possible trend in the future could be projected.
7 Granger and Newbold (1974) call spurious regression has high 𝑅! and t-statistic appear to be significant, but the results are without any economic meaning.
Figure 3-1 Process of Research Method
3.1.1 Structural Change
Since the stability of the regression coefficients is such an important part of the assumptions underlying the flowing regression model, it may be advisable to regard it as a hypothesis to be tested, especially our study period were 15 years. There might occurrence of various factors affecting the process of data generated, for example, the financial crisis, the stock market crash, new technologies to produce, preferences change, or some policy shocks disrupt economic development, it is necessary to confirm whether there is structural changes during the research period. In addition, structural breaks will bias the Unit Root Test statistics toward the non-rejection of a unit root ( Perron’s (2005)).
Research and discussion of structural change in the past (Greene (1993); Kmenta (1986)), the relevant literature on structural econometric model of change considerations, called the stability of regression coefficients (stability or constancy) problem. In this study, we apply the Cumulative Sum of the recursive residuals (CUSUM test), the model estimated the period of sample, if there are structural changes in the value of the forward prediction will growing,
namely the use of recursive residuals of logic to test whether there is a structural change.
The mathematical expression of CUSUM can be expressed as:
-3𝜃 𝑇 − 𝑛<𝑊!<3𝜃 𝑇 − 𝑛 (3.1) In this model, say a variable DGP is AR (1) (3.2a), with the estimated value of the DGP model to predicts the next period, and calculate the error between the predicted value and the actual value (𝑒!)(3.2b), which is the recursive residuals. And 𝜎!,!(3.2c) to represents residual variance of the predicted value. Show define w!(3.2d) as divisor of (3.2b) between (3.2c):
y!=a!+a!y!!!+e! (3.2a) 𝑒!=𝑦!− 𝑦!，t=n+1,n+2,….T (3.2b) 𝜎!,!=Var(𝑒! ↿ 𝑦!!!,𝑦!!!,..,𝑦!) (3.2c)
w
!=
!!!!,!
(3.2d) Kmenta (1986) mentioned in the event of no structural changes, under H!, E(𝑒!)=0 and 𝑒!~(0,𝜎!), w!will be a normal distribution. And define 𝑊! is total
w
!/
𝜎! of n period to t period, when t=n, 𝑊! is between:-𝜃 𝑇 − 𝑛<𝑊!<𝜃 𝑇 − 𝑛 (3.3a) when t=T, 𝑊! is between:
-3𝜃 𝑇 − 𝑛<𝑊!<3𝜃 𝑇 − 𝑛 (3.3b) On the discussion above, where 𝜃=0.948 for a the significance level of 5%, and 𝜃=1.143 for a the significance level of 1%. The null hypothesis is rejected if 𝑊! crosses the boundary associated with the level of significance of the test for some t. If the coefficients are not constant, there may be a tendency of the test for a disproportionate number of recursive residuals to have the same sign and to push 𝑊! across the boundary.
3.1.2 Unit Root Test
In general, many economic or financial time-series variables, have a non-stationary characteristic. The difference is that the mean and variation whether it will change over time.
Stationary series for random external shocks, the impact of the time series data caused only transient impact, and will gradually disappear over time after the time-series data back to the long-term average level of convergence; non-stationary series, the result of any one of random shocks, cause time-series data can not converge to the original equilibrium, the effect for the impact on the data last forever.
Dickey and Fuller (1979) consider three different regression equations that can be used to test for the presence of a unit root, however, in addition to the general nature of the variables themselves may have self-related, but the regression residuals after the estimated correspond with white noise will affect the estimated regression coefficients. If residual not correspond with white noise, which will cause DF value is incorrect. Hence, Dickey and Fuller (1981) proposed ADF regression equation to the right to add a lag period AR (p) conducted a unit root test, known as the ADF test. According to the different characteristics of the variable itself, can distinguish the following three model:
A. Without an intercept term and a trend term：
△ 𝑦!=γ𝑦!!!+ 𝛽!Δ𝑦!!!+𝜀! (3.4a) B. Including an intercept term but not a trend term：
△ 𝑦!=𝑎!+ γ𝑦!!!+ 𝛽!Δ𝑦!!!+𝜀! (3.4b) C. Including an intercept term and a trend term：
△ 𝑦!=𝑎!+ γ𝑦!!!+ 𝑎!𝑡 + 𝛽!Δ𝑦!!!+𝜀! (3.4c) In the model above, △ is first differential operator; 𝑎! is intercept; t is time trend; and γ represent optimal residuals term to correspond with white noise. Hence, we follow the proposal of Enders (2010) and Yang (2011), we estimate a regression equation of the model C first:
The Influence of House prices on Economic Growth and Consumption
Previous studies have found that many economic or financial time-series variables, have a non-stationary characteristic. Granger and Newbold (1974) found that with the non-stationary variables, the estimates of the classic regression model might be spurious regression result. According to the co-integration theory proposed by Engle and Granger (1987), when the cointegration existed between unsteady variables with the same order of difference, their linear combination was a stationary sequence. The regression relationship was still full of economic significance, which could be interpreted that a long-term equilibrium relationship existed between economic variables.
We employ Johansen cointegration test proposed by Johansen (1998), which is based on
Test γ＝0? 𝑦! does not has a unit
VAR approach, uses the maximum likelihood estimation to examine cointegration relationships between the non-stationary time series.
The mathematical expression of
Johansen testcan be expressed as:
△ 𝑦!=𝜋𝑦!!!+ !!!!!!𝜋! △ 𝑦!!!+ 𝛽𝑥!+𝜀! (3.5) where 𝑦! is a vector of non-stationary I(1) series; 𝑥! is a d-vector of exogenous variables;
𝜀! is disturbance vector; !!!!!! 𝜋! is the rank of the long-run impact matrix 𝜋 which equals to the number of cointegrating vectors. Therefore, Johansen proposed the Trace and Maximum Eigenvalue test to determine the number of cointegration.
⋋!"#$%(r)=−𝑇 !!!!!!𝑙𝑛(1-⋋!) (3.6) ⋋!"#(r,r+1)=−𝑇𝑙𝑛(1-⋋!!!) (3.7) where T is the number of observations and ⋋! is the value of characteristic roots. The null hypothesis of the Trace test is H!：rank≤r. For the null hypothesis of the Maximum Eigenvalue test is H!：rank＝r.
3.1.4 Vector Error Correction Model
If there are cointegration between variables in the long-term equilibrium relationship, we can combination of cointegration in the error correction model proposed by Engle and Granger for Vector Error Correction Model. VECM is an appropriate model for a system of cointegrated variables. The mathematical expression of VECM test can be expressed as:
Δy＝a!
+
a!e!!!+ !!!!a!Δ𝑥!!!+ !!!!b!Δy!!!+ε!"(3.8) In this equation, e!!! is measurement deviation for t-1 period long-run equilibrium, is also called the vector of error correction terms; a! is intercept; a! is coefficient of error correction; p is the optimal lag period; 𝜀!" is white noise; ai, bi represent coefficient of short-term dynamic adjustment, can estimate the relationship between the existence of variables and how they affect each other.
3.1.5 Vector Autoregression Model
Vector Autoregression model (VAR) proposed by Sims(1980), the model is appropriate to explore the dynamic interrelationships between the series. And all variables were considered endogenous variables. Each variable was shown by individual lag periods and lag periods of other variables. The model covered all information and could be used to conduct the analysis of direct effects between variables under intertemporal relation. And avoid model identification of the problem. The mathematical expression of VAR can be expressed as:
𝑦!＝α+ !!!!β!𝑦!!!+𝜀! (3.9) In this equation, y! represents the (n×1) vector of endogenous variables like GDP and consumption in this study; y!!! is (n×1) vector composed of ｙ
ｔdeferred i period vector like house prices, stock prices, interest rate, income, and CPI in this study; β! is (n×n) of the coefficient matrix; εｔare the vector of disturbances. Especially, different εｔat the same period can be interrelated but would not related to own lag value and the variables on the right side of the equation.
3.1.6 The Impulse Response Function
Just as an autoregression has a moving-average representation, a VAR can be written as a vector moving average (VMA). The VMA representation is an essential feature of Sims’s (1980) methodology in that it allows us to trace out the time path of the various shocks on the variables contained in the VAR system. The VAR model use lag operator can represent as 𝑌!
＝ Φ!+ Φ!𝑌!!!+𝜀!, and use AR(1) perform MA(∞) as the following：
𝑌!＝𝜇＋𝜀!＋ Ψ!𝜀!!!+ Ψ!𝜀!!!+ Ψ!𝜀!!!+… (3.10) 𝜀!!! can interpretation for t-i period of the accident change or unexpected shock, so after differential 𝜀!!!, we can trace out the lag period effects of one-unit shocks to current period on the time paths of the variable sequences:
Ψ!＝! !!
!!!!! (3.11)
Through the impulse responds function, the analysis observed the impulse of specific variables to other variables. The impulse responds functions could be divided into impulse responds function decomposed by Cholesky and general impulse function. The former needed to set up the order of influences based on the degree of influence of variables, while the latter, proposed by Pesaran and Shin(1988), could be used to analyze results of impulse responses without order, which could prevent the possible distortion of causality caused by preconceptions. The study conducted the analysis based on general impulse responds function. The definition of the function was as below:
GIRF(𝑥!; 𝑢!"#, 𝑛)=E 𝑥!!! 𝑢!"# = 𝜎!,!, Ω!!! - E 𝑥!!! Ω!!! (3.12) In this function, Ω!!! was the information set in t-1 period; 𝜎!,! represented the variance in the j equation in the ith variable-diagonal elements of covariance matrix; n was the length of forecast period. Which measures the effect of one standard error shock to the jth equation at time t on expected values of x at time t+n.