Time Series Assignment Help

Question 1

The dependent variable in this study is the Gross Domestic Product (GDP) while the independent variables are real interest rate and unemployment rate. The main aim of this project work is to determine the type of relationship that exists between the country Gdp and the predictors variables (Unemployment rate and Interest rate), how best can they been used to predict the type of relationship that exist between them is it linear or non-linear. Also through this project we will be able to determine the strength of association between the predictor variables and the country Gdp per capita are they positively correlated or negative in nature and lastly we will be interested also in determining the significance of each variable on GDP are they significant or not and what percentage of variations in GDP can be explained by both independent variables.

Question 2

Fig 1: Time series plot of all original variables

The time series graph above for the levels of all variables shows an evidence of unit roots in all the three variables, while drifts was present in the unemployment rate. The lowest interest rate was recorded in the year 1995, Unemployment rate was at its peak between 1985 and 1987 and lastly the GDP rate fluctuates and was at its peak between 1996 and 1998.

Fig 2: Time series plot of the first differenced variables

The time series plot above for the three variables after been differenced shows no presence of drifts neither unit roots which has been removed due to differencing. The interest rate peaked between year 1996 and 1997 while GDP peaked also in the year 2010.

Question 3

Dependent Variable: LOG_GDP  
Method: Least Squares  
Date: 07/17/21   Time: 22:28  
Sample (adjusted): 1 23  
Included observations: 21 after adjustments 
     
     
VariableCoefficientStd. Errort-StatisticProb.  
     
     
C0.7510480.8754300.8579190.4022
LOG_INTEREST_RATE-0.1859760.325466-0.5714140.5748
LOG_UNEMPLOYMENT0.2487770.5276390.4714910.6430
     
     
R-squared0.019564    Mean dependent var1.000152
Adjusted R-squared-0.089374    S.D. dependent var0.610384
S.E. of regression0.637077    Akaike info criterion2.067711
Sum squared resid7.305603    Schwarz criterion2.216928
Log likelihood-18.71096    Hannan-Quinn criter.2.100095
F-statistic0.179587    Durbin-Watson stat1.046077
Prob(F-statistic)0.837094   
     
     
   
   
   
   

The table above shows the regression model in this project and from here we could deduce the significant variable in our model and describe the regression model. The regression model is insignificant with (F=0.1795, p-value = 0.837) with the p-value of the model greater than 0.05 level of significance we establish the fact that the model is insignificant. The coefficient of determination R-square is the amount of variability in the regression model that the independent variables caused by the independent variable in the model. The R-square was computed to be 0.0196 which means 1.96% of the variation in the model can be accounted for by the independent variables. This is a strong evidence to conclude that the independent variables are not explicit enough to explain the regression model.

Question 4

Null Hypothesis: GDP_GROWTH__ANNUAL____NY_GDP_MKTP_KD_ZG has a unit root
Exogenous: Constant 
Lag Length: 0 (Automatic – based on SIC, maxlag=7)
    
    
   t-Statistic
    
    
Augmented Dickey-Fuller test statistic-3.072421
Test critical values:1% level -3.6793223
 5% level -2.9677673
 10% level -2.6229892
    
    
*MacKinnon (1996) one-sided p-values.


Ho: GDP series has unit root

Hi: GDP series does not have unit root

Since the test statistics is much higher than the 1% critical value in the table above. The null hypothesis is not rejected above and we conclude at a very low probability that the GDP series does have unit root presence.

Null Hypothesis: REAL_INTEREST_RATE_____FR_INR_RINR has a unit root
Exogenous: Constant 
Lag Length: 0 (Automatic – based on SIC, maxlag=7)
    
    
   t-Statistic
    
    
Augmented Dickey-Fuller test statistic-2.3652075
Test critical values:1% level -3.6793223
 5% level -2.9677673
 10% level -2.6229892
    
    
*MacKinnon (1996) one-sided p-values.
    


Ho: Interest rate series has unit root

Hi: Interest rate series does not have unit root

Similarly, the test statistics is much higher than the 1% , 5% and 10% critical values in the table above. The null hypothesis is not rejected above and we conclude at a very low probability that the interest rate series does have unit root presence.

Null Hypothesis: UNEMPLOYMENT_RATE has a unit root
Exogenous: Constant  
Lag Length: 1 (Automatic – based on SIC, maxlag=7)
     
     
   t-Statistic  
     
     
Augmented Dickey-Fuller test statistic-2.7358361 
Test critical values:1% level -3.6891937 
 5% level -2.9718531 
 10% level -2.6251209 
     
     
*MacKinnon (1996) one-sided p-values. 
     


Ho: Unemployment rate series has unit root

Hi: Unemployment rate series does not have unit root

Lastly, the test statistics above is much higher than the 1%, 5% and 10% critical values in the table above. The null hypothesis is not rejected above and we conclude at a very low probability that the unemployment rate series does have unit root presence.

Question 5

Date: 07/17/21   Time: 21:49  
Sample (adjusted): 3 23  
Included observations: 15 after adjustments 
Trend assumption: Linear deterministic trend 
Series: LOG_GDP LOG_UNEMPLOYMENT LOG_INTEREST_RATE  
Lags interval (in first differences): 1 to 1 
     
     
     
Unrestricted Cointegration Rank Test (Trace) 
     
     
Hypothesized Trace0.05 
No. of CE(s)EigenvalueStatisticCritical ValueProb.**
     
     
None0.574292420.807846529.79707330.3697449
At most 10.40183597.997805915.49471280.4656525
At most 20.01911180.28945223.84146550.5905695
     
     
 Trace test indicates no cointegration at the 0.05 level
 * denotes rejection of the hypothesis at the 0.05 level
 **MacKinnon-Haug-Michelis (1999) p-values 
     
Unrestricted Cointegration Rank Test (Maximum Eigenvalue)
     
     
Hypothesized Max-Eigen0.05 
No. of CE(s)EigenvalueStatisticCritical ValueProb.**
     
     
None0.574292412.810040521.13161620.469863
At most 10.40183597.708353614.26460010.4091277
At most 20.0191118280.28945223.84146550.5905695
     
     
 Max-eigenvalue test indicates no cointegration at the 0.05 level
 * denotes rejection of the hypothesis at the 0.05 level
 **MacKinnon-Haug-Michelis (1999) p-values 
     

Ho: No co-integration equation

H1: Ho is not true

Reporting for trace statistics, at the None Co-integration equation we do not reject null hypothesis and conclude that there is no co-integration equation there since p-value (0.369) is greater than 0.05 level of significance. Similarly, at most 1 we conclude there is no co-integration since p-value (0.465) is greater than 0.05 level of significance and lastly at  most 2 equation we do not reject null hypothesis also since p-value (0.591) is greater than 0.05 level of significance we conclude that there is no co-integration equation there also.

Secondly, reporting for Max-Eigen statistics, at the None Co-integration equation we do not reject null hypothesis and conclude that there is no co-integration equation there since p-value (0.469) is greater than 0.05 level of significance. Similarly, at most 1 and at most 2 also have p-values greater than 0.05 level of significance and we conclude there is no co-integration in these two equations also.

Question 6

Unit root test in the first difference of all variables

Null Hypothesis: D_GDP has a unit root 
Exogenous: Constant  
Lag Length: 0 (Automatic – based on SIC, maxlag=6)
     
     
   t-Statistic  Prob.*
     
     
Augmented Dickey-Fuller test statistic-5.84519034.3693e-05
Test critical values:1% level -3.6891937 
 5% level -2.9718531 
 10% level -2.6251209 
     
     
*MacKinnon (1996) one-sided p-values. 


Ho: First differenced GDP series has unit root

Hi: First differenced GDP series does not have unit root

The p-value of the test above is lesser than 0.05 level of significance and 0.01 level of significance. In summary at both 1% and 5% level of significance the null hypothesis is rejected and we conclude first difference GDP has no presence of unit root.

Null Hypothesis: D_INTEREST_RATE has a unit root
Exogenous: Constant  
Lag Length: 0 (Automatic – based on SIC, maxlag=6)
     
     
   t-Statistic  Prob.*
     
     
Augmented Dickey-Fuller test statistic-6.24321591.6145e-05
Test critical values:1% level -3.6891937 
 5% level -2.9718531 
 10% level -2.6251209 
     
     
*MacKinnon (1996) one-sided p-values. 


Ho: First differenced interest rate series has unit root

Hi: First differenced interest rate series does not have unit root

Since p-value above is lesser than 0.01 we conclude at all the three critical values we reject null hypothesis and conclude that there is no unit root in the differenced interest rate series.

Null Hypothesis: D_UNEMPLY_RATE has a unit root
Exogenous: Constant  
Lag Length: 0 (Automatic – based on SIC, maxlag=6)
     
     
   t-Statistic  Prob.*
     
     
Augmented Dickey-Fuller test statistic-2.745617 0.0792
Test critical values:1% level -3.689194 
 5% level -2.971853 
 10% level -2.625121 
     
     
*MacKinnon (1996) one-sided p-values. 
     


Ho: First differenced unemployment rate series has unit root

Hi: First differenced unemployment rate series does not have unit root

Since p-value above is greater than 0.05 we conclude at all the three critical values we do not reject null hypothesis and conclude that there is a unit root in the differenced unemployment rate series.

Regression Model for the differenced data

Dependent Variable: D_GDP  
Method: Least Squares  
Date: 07/17/21   Time: 22:23  
Sample (adjusted): 2 30  
Included observations: 29 after adjustments 
     
     
VariableCoefficientStd. Errort-StatisticProb.  
     
     
C-0.0416210.381910-0.1089800.9141
D_INTEREST_RATE0.1298850.1600400.8115790.4244
D_UNEMPLY_RATE-0.1690290.435213-0.3883810.7009
     
     
R-squared0.036167    Mean dependent var-0.044300
Adjusted R-squared-0.037974    S.D. dependent var1.950834
S.E. of regression1.987529    Akaike info criterion4.309358
Sum squared resid102.7070    Schwarz criterion4.450803
Log likelihood-59.48570    Hannan-Quinn criter.4.353657
F-statistic0.487815    Durbin-Watson stat2.155191
Prob(F-statistic)0.619474   
     
     


Considering the regression model involving the first differenced dependent variable (GDP) and the independent variables interest rate and unemployment rate which are differenced at first levels also. The regression model is insignificant with (F=0.4878, p-value = 0.619) with the p-value of the model greater than 0.05 level of significance we establish the fact that the model is insignificant. The coefficient of determination R-square is the amount of variability in the regression model that the independent variables caused by the independent variable in the model. The R-square was computed to be 0.0361 which means 3.61% of the variation in the model can be accounted for by the independent variables. This is a strong evidence to conclude that the independent variables are not explicit enough to explain the regression model.

Question 7

The First model with the log dependent variable is not significant at 0.05% level of significance since the p-value of the model is greater than 0.05 level of significance. While, 1.96% of the variation in the model were determined by both interest rate and unemployment which were also logged in the model. Similarly, the interest rate has a negative insignificant effect on the dependent variable (GDP) in this model and unemployment rate as a positive insignificant effect on GDP the dependent variable in this model. Lastly, unit root were present in the three series used in this model.

The second model with differenced dependent variable and differenced independent variables shows that the model is not significant here also since the p-value of the model is greater than 0.05% level of significance. 3.61% of the variation in this model was explained by the independent variables. Both unemployment rate and interest rate have a positive insignificance effect on the dependent variable (GDP) in this model and lastly, both GDP and Interest rate has no unit root after differenced but unemployment rate does have unit root presence still after differencing.

Overall best model is considered as model 2 with difference dependent variable GDP and differenced independent variables since the model has the higher coefficient of determination R-squared. So we can conclude model 2 is better than Model 1.

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