Adf And Kpss Tests Economics Essay
Before presenting the results, it is importance to check for the time series data existence of stationary property of each variable whether the variables under consideration are stationary in level form. It is important to ensure that the variables used in the regression are not subject to spurious correlation. We are using unit root tests to investigate the stationary status of each variable by Augmented Dickey-Fuller (ADF) and Kwiatkowski, Phillips, Schmidt and Shin (KPSS). These two tests will test for with and without time trend at level form and indicate lag lengths based on the Akaike Information Criterion (AIC). Table 1 is the result for ADF while Table 2 is KPSS.
Table 1 show that CPI and RGFCF, both are significant at 5% level at the level form without linear trend; significant at 1% level for the level form with linear trend. RGDP is significant at 10% level for the level form without trend and significant at 5% level with linear trend. While for the POP, it cannot significant at level form without linear trend. However, POP significant at 5% level for the level form with linear trend.
Table 2 shows that all variables are significant at 1% level for both with and without linear trend at level form. Which is explained that the t-statistic have sufficient evidence do not reject the null hypothesis at level form in both with and without linear trend. These mean that the series are stationary at level form.
In short, all the variables show stationary at the level form, I(0). Which it obeys the theory or concept of stationarity of financial data where it is predicted to be non-stationary at level form and stationary after the first difference. Thus this allows us to proceed to the cointegration tests.
Variables
Augmented Dickey-Fuller
level
Constant without linear trend
Constant with linear trend
CPI
-3.042517 ** (3)
-6.276123 *** (7)
RGFCF
-3.393829 ** (1)
-12.31312 *** (1)
RGDP
-2.632101 * (4)
-4.034734 ** (3)
POP
-0.659324 (1)
-4.187427 ** (6)
Table 1: Results of Unit Root tests with ADF
Notes: figures within parentheses indicate lag lengths. Lag length for ADF tests have been decided on the basis of Akaike Information Criterion (AIC) (Akaike, 1974). The ADF tests are based on the null hypothesis of unit roots. ***, **, and * indicate significant at 1%, 5% and 10% levels respectively, based on the critical t statistics as computed by Mackinnon (1996).
Table 2: Results of Unit Roots tests with KPSS
Variables
Kwiatkowski, Phillips, Schmidt and Shin (KPSS)
level
Constant without linear trend
Constant with linear trend
CPI
0.170339 ***
0.172165 ***
RGFCF
0.686537 ***
0.086532 ***
RGDP
0.120369 ***
0.065365 ***
POP
0.491314 ***
0.117854 ***
Notes: figures within parentheses indicate lag lengths. Lag length for ADF tests have been decided on the basis of Kwiatkowski, Phillips, Schmidt and Shin (KPSS). The KPSS tests are based on the null hypothesis of unit roots. ***, **, and * indicate significant at 1%, 5% and 10% levels respectively, based on the critical t statistics as computed by Mackinnon (1996).
4.2 Granger Causality Tests
After identify ADP and KPSS test to ensure all the variables are stationary, we used Granger Causality test to estimate the linear causation between inflation and economic growth and the results are shown in Table 3.
Table 3: Pair wise Granger Causality Tests
Sample: 1960 – 2005
Lags: 1
 Null Hypothesis:
Obs
F-Statistic
Prob.Â
 CPI does not Granger Cause RGDP
 44
 4.68436
0.0363
 RGDP does not Granger Cause CPI
 21.2996
4.E-05
Both the null hypothesis is rejected at 1-5 percent level of significance, which implies that inflation rate does Granger Causality real GDP growth and real GDP growth does Granger Causality inflation rate. This test statistic shows that the causality between two variables is bi-directed. The variables are co-integrated because there is a long-run relationship between inflation rate and real GDP growth. We further applied the Akaike Information Criterion (AIC) and Schwarz Information Criterion (SIC) to define the lag length used for inflation rate and real GDP growth. The result is important to identify the choice of dependent and independent variable for the threshold model specification. In addition, the inflation rate is causing growth at lag one (lag=1) for the period from 1961 to 2005. Hence, we generate the equation by adding lag one for the inflation rate in the estimation model.
4.3 Threshold Model estimation
We run the data using ordinary least squares (OLS) econometric technique and the inflation rate are kept at lag one after estimating for Granger Causality test. The optimal threshold level is the minimum value of RSS (residual sum of squares) as shown in table 4. Table 4 also illustrates the result of t-statistics and P-values for the estimation equation. From the result, it shows that both 3 and 5 percent inflation level also get the minimum value of RSS. This means that the optimal level of threshold is about 3 to 5 percent. In order to find out the threshold level of inflation, we further specified the value of k from the range of 3 to 5 percent into percentage point and the estimation value is shown as Table 5. The results in Table 5 indicated that 3.1 percent inflation level is the optimal level of threshold in which the value of k is the one with minimizes the residual sum of squares (RSS).
4.3 .1 Estimation of OLS regression
Table 4: Estimation of OLS regression at k = 1 to 5%
(Dependent Variable: real GDP growth)
K (%)
Variable
Coefficient
Std. Error
t-statistics
P- value
RSS
1
Inflation
Inflation (-1)
(inf>1)*(inf-1)
Investment growth
Population growth
C
1.538332
-0.564809
-0.976010
0.170379
-1.939174
9.314812
0.583956
0.298513
0.663202
0.035571
0.444884
1.162268
2.634331
-1.892076
-1.471665
4.789804
-4.358832
8.014344
0.0121
0.0661
0.0000
0.0001
0.1493
0.0000
7.223265
2
Inflation
Inflation (-1)
(inf>2)*(inf-2)
Investment growth
Population growth
C
0.969290
-0.582696
-0.420789
0.175398
-1.982495
9.648625
0.336110
0.302481
0.337848
0.040910
0.459113
1.116383
2.883850
-1.926386
-1.245497
4.287397
-4.318101
8.642757
0.0064
0.0616
0.2206
0.0001
0.0001
0.0000
7.335498
3
Inflation
Inflation (-1)
(inf>3)*(inf-3)
Investment growth
Population growth
C
0.826579
-0.599289
-0.359792
0.188432
-2.344833
10.63177
0.304288
0.283725
0.202986
0.039605
0.522635
1.144140
2.716439
-2.112214
-1.772499
4.757815
-4.486557
9.292370
0.0099
0.0413
0.0843
0.0000
0.0001
0.0000
7.051917
4
Inflation
Inflation (-1)
(inf>4)*(inf-4)
Investment growth
Population growth
C
0.811575
-0.749978
0.113362
0.128502
-1.700261
9.726264
0.315729
0.291090
0.257392
0.038080
0.532298
1.201991
2.570482
-2.576447
0.440428
3.374569
-3.194193
8.091795
0.0142
0.0140
0.6621
0.0017
0.0028
0.0000
7.596177
5
Inflation
Inflation (-1)
(inf>5)*(inf-5)
Investment growth
Population growth
C
0.826574
-0.789422
0.821631
0.110625
-1.472938
9.327557
0.304094
0.277041
0.459772
0.032438
0.475637
1.123008
2.718158
-2.849476
1.787040
3.410344
-3.096766
8.305869
0.0098
0.0070
0.0819
0.0016
0.0037
0.0000
7.043056
*(inf>k)*(inf-k) denotes the dummy variable
Table 5: Estimation of OLS regression at k = 3 to 5%
(Dependent Variable: real GDP growth)
K (%)
Variable
Coefficient
Std. Error
t-statistics
P- value
RSS
3
Inflation
Inflation (-1)
(inf>3)*(inf-3)
Investment growth
Population growth
C
0.826579
-0.599289
-0.359792
0.188432
-2.344833
10.63177
0.304288
0.283725
0.202986
0.039605
0.522635
1.144140
2.716439
-2.112214
-1.772499
4.757815
-4.486557
9.292370
0.0099
0.0413
0.0843
0.0000
0.0001
0.0000
7.051917
3.1
Inflation
Inflation (-1)
(inf>3.1)*(inf-3.1)
Investment growth
Population growth
C
0.819686
-0.599990
-0.365668
0.189027
-2.365807
10.68969
0.303305
0.282258
0.198960
0.039084
0.522171
1.147435
2.702515
-2.125684
-1.837890
4.836480
-4.530714
9.316161
0.0102
0.0401
0.0739
0.0000
0.0001
0.0000
7.011681
3.2
Inflation
Inflation (-1)
(inf>3.2)*(inf-3.2)
Investment growth
Population growth
C
0.813290
-0.609849
-0.346033
0.185869
-2.341698
10.67837
0.304849
0.283432
0.201610
0.039245
0.527866
1.161110
2.667843
-2.151662
-1.716349
4.736069
-4.436160
9.196697
0.0112
0.0378
0.0942
0.0000
0.0001
0.0000
7.085655
3.3
Inflation
Inflation (-1)
(inf>3.3)*(inf-3.3)
Investment growth
Population growth
C
0.808974
-0.630219
-0.292763
0.177525
-2.253800
10.55958
0.308498
0.286697
0.208103
0.039526
0.534224
1.177629
2.622301
-2.198207
-1.406819
4.491332
-4.218832
8.966817
0.0125
0.0341
0.1676
0.0001
0.0001
0.0000
7.256989
3.4
Inflation
Inflation (-1)
(inf>3.4)*(inf-3.4)
Investment growth
Population growth
C
0.805122
-0.650546
-0.232901
0.168643
-2.156660
10.42119
0.311662
0.289750
0.214940
0.039689
0.538534
1.190453
2.583316
-2.245200
-1.083561
4.249078
-4.004684
8.753971
0.0138
0.0307
0.2854
0.0001
0.0003
0.0000
7.406122
3.6
Inflation
Inflation (-1)
(inf>3.6)*(inf-3.6)
Investment growth
Population growth
C
0.807381
-0.691159
-0.122278
0.153417
-1.988913
10.17585
0.315266
0.291690
0.230561
0.039680
0.543951
1.207686
2.560950
-2.369499
-0.530351
3.866320
-3.656419
8.425900
0.0145
0.0230
0.5990
0.0004
0.0008
0.0000
7.578855
3.7
Inflation
Inflation (-1)
(inf>3.7)*(inf-3.7)
Investment growth
Population growth
C
0.807354
-0.706396
-0.069755
0.146940
-1.915185
10.06400
0.316081
0.292153
0.238509
0.039497
0.543696
1.210660
2.554263
-2.417897
-0.292465
3.720337
-3.522527
8.312827
0.0148
0.0205
0.7715
0.0006
0.0011
0.0000
7.617805
3.8
Inflation
Inflation (-1)
(inf>3.8)*(inf-3.8)
Investment growth
Population growth
C
0.807750
-0.720644
-0.015039
0.140795
-1.844324
9.954601
0.316440
0.292325
0.246159
0.039191
0.541739
1.210691
2.552617
-2.465216
-0.061095
3.592527
-3.404451
8.222247
0.0148
0.0183
0.9516
0.0009
0.0016
0.0000
7.634203
3.9
Inflation
Inflation (-1)
(inf>3.9)*(inf-3.9)
Investment growth
Population growth
C
0.809120
-0.735339
0.046430
0.134571
-1.771822
9.840773
0.316347
0.291994
0.252579
0.038722
0.537991
1.207920
2.557700
-2.518335
0.183822
3.475323
-3.293408
8.146872
0.0146
0.0161
0.8551
0.0013
0.0021
0.0000
7.628169
4
Inflation
Inflation (-1)
(inf>4)*(inf-4)
Investment growth
Population growth
C
0.811575
-0.749978
0.113362
0.128502
-1.700261
9.726264
0.315729
0.291090
0.257392
0.038080
0.532298
1.201991
2.570482
-2.576447
0.440428
3.374569
-3.194193
8.091795
0.0142
0.0140
0.6621
0.0017
0.0028
0.0000
7.596177
4.1
Inflation
Inflation (-1)
(inf>4.1)*(inf-4.1)
Investment growth
Population growth
C
0.813492
-0.760287
0.170993
0.124043
-1.646754
9.638877
0.314833
0.289823
0.265606
0.037537
0.527469
1.196228
2.583886
-2.623281
0.643784
3.304522
3.121994
8.057725
0.0137
0.0125
0.5236
0.0021
0.0034
0.0000
7.552578
4.2
Inflation
Inflation (-1)
(inf>4.2)*(inf-4.2)
Investment growth
Population growth
C
0.814782
-0.767124
0.220848
0.120950
-1.609142
9.576530
0.313960
0.288694
0.278565
0.037145
0.523851
1.191449
2.595175
-2.657221
0.792806
3.256148
-3.071754
8.037718
0.0134
0.0115
0.4328
0.0024
0.0039
0.0000
7.510721
4.3
Inflation
Inflation (-1)
(inf>4.3)*(inf-4.3)
Investment growth
Population growth
C
0.816150
-0.773634
0.278128
0.117888
-1.571341
9.512792
0.312816
0.287248
0.291258
0.036629
0.518916
1.184728
2.609045
-2.693259
0.954918
3.218462
-3.028123
8.029515
0.0129
0.0105
0.3457
0.0026
0.0044
0.0000
7.456034
4.4
Inflation
Inflation (-1)
(inf>4.4)*(inf-4.4)
Investment growth
Population growth
C
0.817551
-0.779375
0.342362
0.115024
-1.535224
9.450467
0.311364
0.285454
0.303176
0.035972
0.512475
1.175766
2.625705
-2.730304
1.129254
3.197581
-2.995704
8.037712
0.0124
0.0095
0.2659
0.0028
0.0048
0.0000
7.387056
4.5
Inflation
Inflation (-1)
(inf>4.5)*(inf-4.5)
Investment growth
Population growth
C
0.819140
-0.783980
0.411602
0.112607
-1.503814
9.394421
0.309627
0.283369
0.314048
0.035185
0.504553
1.164552
2.645568
-2.766638
1.310632
3.200417
-2.980485
8.066984
0.0118
0.0087
0.1978
0.0028
0.0050
0.0000
7.304748
4.6
Inflation
Inflation (-1)
(inf>4.6)*(inf-4.6)
Investment growth
Population growth
C
0.826182
-0.791957
0.475658
0.111246
-1.488082
9.365990
0.308440
0.282444
0.331298
0.034615
0.497872
1.154816
2.678578
-2.803944
1.435742
3.213786
-2.988884
8.110371
0.0109
0.0079
0.1593
0.0027
0.0049
0.0000
7.242096
4.7
Inflation
Inflation (-1)
(inf>4.7)*(inf-4.7)
Investment growth
Population growth
C
0.831194
-0.797041
0.545831
0.110370
-1.477221
9.344919
0.307201
0.281199
0.351402
0.034007
0.491151
1.145051
2.705703
-2.834436
1.553297
3.245510
-3.007671
8.161138
0.0101
0.0073
0.1286
0.0024
0.0047
0.0000
7.179129
4.8
Inflation
Inflation (-1)
(inf>4.8)*(inf-4.8)
Investment growth
Population growth
C
0.827864
-0.793620
0.626005
0.110009
-1.469475
9.328254
0.305943
0.279437
0.380078
0.033468
0.486068
1.137663
2.705945
-2.840064
1.647046
3.286991
-3.023188
8.199489
0.0101
0.0072
0.1078
0.0022
0.0045
0.0000
7.126222
4.9
Inflation
Inflation (-1)
(inf>4.9)*(inf-4.9)
Investment growth
Population growth
C
0.823792
-0.788270
0.715461
0.110231
-1.468281
9.321785
0.304688
0.277659
0.411601
0.032828
0.479930
1.129002
2.703724
-2.838987
1.738240
3.357893
-3.059364
8.256657
0.0102
0.0072
0.0903
0.0018
0.0041
0.0000
7.072592
5
Inflation
Inflation (-1)
(inf>5)*(inf-5)
Investment growth
Population growth
C
0.826574
-0.789422
0.821631
0.110625
-1.472938
9.327557
0.304094
0.277041
0.459772
0.032438
0.475637
1.123008
2.718158
-2.849476
1.787040
3.410344
-3.096766
8.305869
0.0098
0.0070
0.0819
0.0016
0.0037
0.0000
7.043056
*(inf>k)*(inf-k) denotes the dummy variable
The estimation outputs for the model are generated as shown below:
GROWTH t = 10.6897 + 0.8197INF t – 0.56INF t -1- 0.3657DUMMY
– 2.3658POP t + 0.189INV t
t-stat = (9.3162) (2.7025) (-2.1257) (-1.8379)
(-4.5307) (4.8365)
R-squared = 0.7391
F-stats = 21.5272 ; Prob (F-statistic) = 0.0000
S.E. of regression = 0.4296; Mean dependent variable = 6.7262
Ceteris paribus, equals to 0.8197 means that when inflation rate rises by 1 percent, real GDP growth rate will increase by 0.82 percent on average. is 0.56, which indicates that when previous year inflation rate rises by 1 percent, the real GDP growth rate will decrease by 0.56 percent on average. is – 0.3657, indicates that when the threshold level of inflation increases by 1 percent, the real GDP growth rate will decrease by 0.37 percent on average. is -2.3658, indicating that when population growth increases by 1 percent, the real GDP growth rate will decrease by 2.37 percent on average. is 0.189 means that for every 1 percent increase in real investment, the real GDP growth rate will be increased by 0.19 percent on average.
4.3.2 Test of significance
Table 6
T-test:
Coefficient
β1 (inf)
β2 (inf(–1))
β3 dummy
β4 pop
β5 inv
Exp. Sign
positive
negative
negative
negative
positive
t-stat
2.7025
-2.1257
-1.8379
-4.5307
4.8365
P-value
0.0102**
0.0401**
0.0739*
0.0001***
0.0000***
Decision
Significant
Significant
Significant
Significant
Significant
The T- tests are based on the null hypothesis. ***, **, and * indicate significant at 1%, 5% and 10% levels respectively, based on the critical t statistics as computed by Mackinnon (1996).
H0: βi = 0
H1: βi ≠0, i = 1, 2, 3, 4, 5
Decision rule: Reject H0 if P-value is smaller than α = 0.1, otherwise do not reject H0.
Conclusion: There is sufficient evidence to conclude that all variables are significantly affecting real GDP at α = 0.1.
F-test:
H0: β1 = β2 = β3 = β4= β5 = 0;
H1: At least one β is ≠0
Decision rule: Reject H0 if Prob(F-stat) smaller than α = 0.1, otherwise do not reject H0
Decision: Since Prob (F-stat) is 0.0000 which is smaller than 0.1, thus we reject H0
Conclusion: There is sufficient evidence to prove that there is at least one of the explanatory variables is significantly affecting the economic growth in Malaysia.
Goodness of fit:
= 0.7391, which indicates that there is 73.91 percent of the variation in real GDP of Malaysia that can be explained by the variations of inflation rate, previous year inflation rate, threshold level of inflation, population growth and real investment.
Standard error-to-mean ratio:
Standard error-to-mean ratio =(S.E. of regression / Mean dependent variable) *100%
= (0.4296/ 6.7262) *100%
= 6.39%
The standard error of regression is 6.39%, which is considered very small, indicates
that the estimates value is close to the true value. Hence, this model is a good fit.
4.4 Diagnostic Checking
We show the optimal level of inflation for the diagnostic checking and it illustrated in Table 6. For diagnostic testing, the problem such as multicolinearity, autocorrelation, heteroskedasticity, model misspecification error and normality test for residual are included. The result is shown as below:
Table 6: Diagnostic tests for equation at k=3.1%
Test
Hypothesis
Statistics
Result
Multicollinearity
VIF= 1/ (1-R2) = 1/ (1-0.4838) =1.9372
Not serious multi. problem
Autocorrelation
H0: There is no autocorrelation problem
Prob.Chi-Square(2) = 0.0000
reject H0
H1: There is autocorrelation problem
Heteroskedasti
-city
H0: There is no heteroskedasticity problem
Prob. F(1,41) = 0.0000
reject H0
H1: There is heteroskedasticity problem
Misspecification test
H0: There is no misspecification error
Prob. F(1,37) = 0.7274
Do not reject H0
H1: There is misspecification error
Note: ** denote significant level at 5percent
Normally distribution:
Figure 1:
For normality test for residual, Figure 1 shows that the P-value of JB-stat test is 0.423364, which is higher than 5% significant level. Thus we do not reject null hypothesis at 5% significant level and conclude that the residual are normally distributed.
For multicollinearity test, we perform regression analysis for the highly correlated pair of independent variables to get the R squared and calculate the VIF. Table 7 is the correlation analysis for every pair of independent variables.
Table 7: Correlation checking
Â
RGDP
CPI
RGFCF
POP
_RGDP
1
0.529119
0.517047
-0.07067
CPI
0.529119
1
0.108155
-0.39804
RGFCF
0.517047
0.108155
1
0.695556
_POP
-0.07067
-0.39804
0.695556
1
From the table above, the result shows that population and real investment are highly correlated which is about 0.695556.
POP = 2.1984+0.04063RGFCF
R-squared = 0.4838
VIF(Variance Inflation Factor )= 1.9372
VIF is 1.9372 which is less than 10, this claim that there is not a serious multicollinearity problem between Population and real investment. Hence, we can leave the model alone if the VIF is not so serious and t-stat is statistically significant.
We use Breusch-Godfrey Serial Correlation LM Test to examine the existence of autocorrelation. From the result showed above, the P-value of the Chi-square test is 0.0000 which is smaller than 5% significant level. Thus we reject null hypothesis at 5% significant level since there is sufficient evidence to conclude that there is first order autocorrelation problem in the model.
The motive of running the ARCH Test is to examine the existence of heteroscedasticity. From the result showed above, the P-value of F-stat test is 0.0000 which is smaller than 5% significant level. Hence, we reject null hypothesis at 5% significant level since there is sufficient evidence to conclude that it is heteroskedasticity problem in the model.
The purpose for Ramsey Reset Test is to test the misspecification error in the model. Based on the output above, the P-value of F-stat test is 0.7274 and it is more than 5% significant level. Therefore, we do not reject null hypothesis at 5% significant level since there is insufficient evidence to prove that there is misspecification error in the model.
The results above indicate that there is an autocorrelation and heteroskedasticity problem take place in the estimated equation. Therefore, we applied White’s procedure to solve this problem and the result is shows in Table 8. We comparing the output with regular OLS output to check whether heteroskedasticity is a serious problem in the model. The White’s Heteroscedasticity-Consistent Variance and standard errors, also known as robust standard errors can be implemented so as to asymptotically valid statistical illation can be made about the true parameter values (Gujarti). Furthermore, we examined White’s Heteroscedasticity-Consistent Variance and standard errors along with the OLS variances and standard errors.
Table 8: White Test at k=3.1%
K (%)
Variable
Coefficient
Std. Error
t-statistics
P- value
RSS
3.1
Inflation
Inflation (-1)
(inf>1)*(inf-1)
Investment growth
Population growth
C
0.819686
-0.599990
-0.365668
0.189027
-2.365807
10.68969
0.331024
0.307544
0.192164
0.029116
0.342021
0.677976
2.476211
-1.950911
-1.902896
6.492182
-6.917141
15.76707
0.0178
0.0585
0.0647
0.0000
0.0000
0.0000
7.011681
GROWTH t = 10.6897 + 0.8197INF t – 0.56INF t -1- 0.3657DUMMY
– 2.3658POP t + 0.189INV t
OLS se = (1.1474) (0.3303) (0.2823) (0.199)
(0.5222) (0.039)
t-stat = (9.3162) (2.7025) (-2.1257) (-1.8379)
(-4.5307) (4.8365)
White se = (0.678) (0.331) (0.3075) (0.1922)
(0.342) (0.0291)
t-stat = (15.767) (2.4762) (-1.951) (-1.9029)
(-6.9171) (6.4922)
The preceding results show that White’s Heteroscedasticity-Consistent Variance and standard errors are considerable larger for inflation rate and previous year inflation rate, therefore the estimated t values are much smaller than those obtained by OLS. However, the standard errors for threshold level of inflation, population growth and real investment are smaller than OLS output, thus the estimated t values are larger than OLS output. Besides, both estimators are statistically significant at the 10 percent level.
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