Q1 Solution
We have referred the data of a time series starting from 1970 upto the year 2018 for China using reference 1.
The imports, and exports of China is compared to see its effect on annual GDP.
A study is made to see whether any formulation is inferred by utilizing regression models.
The variables choice are made on the model to arrive at an economic equation. Finally a linear model is established.
It is known that the GDP of a country is greatly depended on imports and exports.
The imports are taken as a negative values. It is seen that imports corroborates linearly but the relationship is inversely proportional.
GDP is determined by the amount of exports and imports. So GDP becomes a dependent variable while Imports, exports as 2 independent variables. As GDP is greatly affected by imports and experts, we expect a good correlation. This will provide the good validation for the model.
Q2 Solution
Here Ln( GDP ) is plotted as seen in Figure 1, using the time series. The observed trend line has a rising trend with a slow rate. We observe some reduced slope from year 1998 up to 2009. We observe some fluctuation but the overall trend line is a linear line. With a linear trend it can be concluded that the time series is does not show stationary behavior. The variance, mean both change when ln(GDP) is plotted with time for the period of data considered. Data shows white noise superimposed on some drift.
Figure 1 : Plot of Ln( GDP ) with time
Next we study the plot of LN( E ) with time seen in Figure 2, using the time series data. Here we see an increasing trend. The value of Ln€ remains confined within 24$  31$ with some minor fluctuates superimposed on the line with marginal slope thus showup as liner. This slowly rising trend is main trend this time series is also non_stationary. The mean is liner, variance is aslo linear. It is showing deterministic trend.
Figure 2 : Plot of Ln(E) with time
Next we study the plot of LN( I ) with time seen in Figure 3, the Ln(I) data series shows decreasing behavior whichvaries within a narrow range of 20$ to 30$ . Here also we see fluctuating behaviour superimposed over the linear trend line. A linear befavior is seen in this data set and the data set show up a non_stationary tend line. The variance, mean also follow the linear trend. It shows drift and a white noise along with a deterministic trendline.
Figure 3 : Plot of Ln(I) with time
Once the mean is removed the first difference in the Ln( DGP ) is plotted in figure 4, as a time series.
It is seen that the trend has disappeared and no clear pattern is possible to arrive at from the graph. Hence this can be approximated as a time series with stationary behaviour having constant variance, mean over the studied time interval.
Figure 4 : Plot of Diff [ Ln(GDP)] with time
Similarly, we make the plot of diff( Ln( E )) which is the first difference of Ln( E ) and diff( Ln( I )) which is the first difference of Ln( I ) as shown in Figures 5 and 6.
We observe that the diffLn( GDP ) shows a fluctuating pattern with mean close to 0.8 It does not show any visible trend so this is a stationary series with time on the x axis having a constant variance, mean plotted over the intervals of times. Also, Ln( E ) fluctuates around mean. The mean value is close to 0.1.Thus we have nonzero average. For ln( I ) around has fluctuating behaviour with mean close to a value of 0.2. Thus LN(I) has non_zero mean. But We can see that a non_stationary processcan be made stationary process.
Figure 5 : Plot of Diff [ Ln( E ) ] with time
Figure 6 : Plot of Diff [ Ln( I ) ] with time
Question 3:
The following regression coefficients are obtained using process of regression.

Coefficient of regression 
Intercept 
9.7 
Ln( E ) 
.52 
Ln( I ) 
.18 
Error 
.30 
Ln( GDPt ) = 9.73 + 0.52 * ln( Et )  0.18 * ln( It ) + 0.30
Starting GDP for year 1970 is which we get from the intercept. The graph is Ln ( Et ) vs time and it gives an increase in Ln( GDP ) by 0.52 for every unit increase in Ln( Et ). It is essential to keep Ln ( l ) constant. The coefficient of Ln ( l ) shows that Ln( GDP ) decreases by 0.18 for a unit change in Ln(lt ). Standard error here, in 0.30 .
Question 4:
To start with we apply the test given by Durbin Watson for checking the validity or spuriousness of the regression.
RSquare 
0.97 
DurbinWatson 
0.10 
Since the value from R_square test has a vvalue greter then coefficient from test done using Durbin Watson. The regression seen is not believable got from regression model using linear trend line. But even a high R_square value is also confirming the not stationalty, since the series being analysed is a nonstationary series.
Now testing for all the three_time series Ln(GDP), Ln ( E ) and Ln ( I ), is done to investigate the existence of unit roots. In this work we use “ Augmented_Dickey Fuller_test ( reference 3) number three “ (“ such tests are used to explore whether any trend exits or not no trend is there , by trend we mean either sloping up or sloping down behaviour. The trend line can also be like slowly turning around, this also needs to be explored from the selected data (Reference 2) for Ln( GDP ). We can performing regressional analysis using first difference method and taking the “LAG” lag , this is defined as first difference as shown below:
DLn( GDP t )= m + b * lagLn( GDPt1 )+ lt + et
Here m is the drift component
The lt is the deterministic trend
The et =is the white_noise which is 5%, and the critical value is equal to a value of negative four point two.

The _coefficient 
The_Standard Error 
The_T Statistics 
The_Intercept 
.19 
.234 
.817 
lagLn( GDP ) 
.01 
.009 
+1.182 
lagLn( GDP ) is independent variable lag of Ln( GDP )
difflagLn( GDP ) is defined as the difference of the independent variable lag of the first difference of Ln( GDP )
We can compute the t statistics using the data of lagLn( E ), the value of lagLn( E ) which is 1.188 to the critical value is 4.22 it is not possible to reject the null hypothesis, thus the data needs to be differenced to make it stationary.
Using Augmented Dickey Fuller test number three for Ln( E )
DLn( Et) = m + blagLn ( Et1 + lt + et
critical value with 5% confidence is 4.22

The _coefficient 
The_Standard Error 
The_T Statistics 
The_Intercept 
.53 
.296 
1.8 
lagLn( E ) 
.012 
.011 
1.4 
lagLn( E ) is the independent variable lag of Ln ( E )
difflagLn( E ) is the lag of independent for the difference of first order on Ln ( E )
Comparing the t_statistics valuefor lagLn ( I ) 1.4 to the critical value 4.22. We cannot reject the nullHypothesis. The data needs to be differenced, thus the data becomes alomost stationary
Using the method of Augmented_Dickey Fuller test number three on Ln( E )
DLn( It ) = m + blagLn( It1 ) + lt + et
Question 5:
Performing OLS_estimate on Ln( GDP ), Ln( E ), Ln( I ) where dependent variable is Ln( GDP ) and Ln( E ) and independent variables are Ln( I ) , Ln( E ).
The_R value 
The_R Square value 
The_Adjusted_R Square 
The_Standard Error of Estimate 
Values using Durbin_Watson 
.98 
0.97 
0.96 
0.30 
0.10 

B 
The_Standard Error 
The beta 
The_T Statistics 
The_Significance values 
The_Intercept 
9.73 
.52 

18.66 
.0 
Ln( E ) 
.52 
.31 
.74 
1.67 
.10 
Ln( I ) 
.18 
.32 
.25 
.55 
.58 
There is no co_integration.
We can also use the Johansen test for cointegration since all the threetime series are nonstationary both type trace and eigen.
Results for type equal to eigen
Type=eigen 
The_T Statistics 
Critical value 5five percent 
H_0= R=0 
Ln ( E ),Ln ( I ) 
13.56 
15.7 
Without co_integration 
Ln (GDP ), Ln ( E ) 
11.89 
15.7 
Without co_integration 
Ln ( GDP ), Ln ( I ) 
11.89 
15.7 
Without co_integration 
Results for type equal to trace
Type=eigen 
The_T Statistics 
Critical value 5five percent 
H_0= R=0 
Ln ( E ),Ln ( I ) 
14.82 
19.9 
Without co_integration 
Ln (GDP ), Ln ( E ) 
16.41 
19.9 
Without co_integration 
Ln ( GDP ), Ln ( I ) 
16.41 
19.9 
Without co_integration 
Question 6:
Here we perform unit_root test from which rising trend is removed, the null hypothesis is also accepted. The data needs one more operation called the difference. This operation makes it stationary data with a modified time series. Now we perform the regression over this first differenced data for each Ln( GDP ), Ln( E ), Ln( I ). Further we use Augmented Dickey Fuller test two. This test can be performed only when we have flat time series, which means that it does not show any trend. Further when it is having very low value of slope then we can perform regression using equation as shown below:
DLn( GDPt ) = m + blagLn( GDPt1 ) + et
Where m is the drift and the et is white noise
For test two we use level of five percent with critical value = 3.0 for the null hypothesis; data becomes stationary when subjected to the difference operation.

Coefficients 
Standard Error 
tStatistics 
Intercept 
.067 
.024 
2.756 
L.lagdiffLn( GDP ) 
.65 
.193 
3.34 
L.lagdiffLn( GDP ) = lag of the lag of first difference of Ln( GDP )
tstatistics prdits Ln( GDP ) = 3.34

Coefficient 
Standard_Error 
t_Statistics 
Intercept 
0.116 
0.037 
3.1 
L.lagdiffLn( GDP ) 
0.809 
0.201 
4.0 
L.lagdiffLn( E ) = lag of the lag of first difference of Ln( E )
DLnEt = m + blagLnEt1 + et

Coefficients 
Standard_Error 
tStatistics 
Intercept 
.161 
.035 
4.64 
L.lagdiffLn( GDP ) 
.647 
.174 
6.26 
L.lagdiffLn( I ) is defined as lag_of the lag of first difference of Ln( I )
DLn( It )= m + blagLn( It1 )+ et for diffLn( I )using ADF test number two as well since it is fluctuating around an average that is nonzero. Since 6.26 is less than 3.0; rejecting null hypothesis for modified time series as well. We can conclude that this series is now stationary.
Using the results got by performing OLS_regression predicts:
R_value 
R_Square value 
The_R Square adjusted value 
The_Standard Error of Estimate 
The_Durbin Watson test result values 
.495 
0.25 
0.076 
0.076 
1.47 

B 
Standard Error 
The_Beta values 
The_t_statistics 
Measure of significance 
predictedConstant 
.06 
.016 

3.72 
.001 
Ln( E ) 
.135 
.102 
.222 
1.33 
.192 
Ln( I ) 
.164 
.085 
.323 
1.92 
.061 
Diff( GDP ) = .06 + .135 * diffLn( E ) .164 * diffLn( I ) + . 075625
Answer Q7 :
Here we prefer to use the 2nd model in comparison to the 1st model for the estimation of data.
The first selected data model has significantly high R_square value of 0.96 which means that independent variables are able to explain results within confidence band of 96% confidence if we need to get the variation prediction as a function of dependent variable. It is very much less likely as Durbin_Watson test values are less than the R_square with value = 0.96. This indicates that we have positive auto_correlation.
When the imports and exports has increasing trend over an year. It is very much unlikely that the imports, exports will increase in the next year given that it has increased in the previous year.
From the regression coefficients of the intercept, Ln( E ) and Ln( I ) , which are statistically significant, we consider the tstatistics. We can notice that the whole regression has been rendered misleading. From the levelled graphs, we can observes that the time series show that it is nonstationary. Nonstationary series have non stationary statistical properties hence they change over time. Thus it is necessary that we change the nonstationary data into stationary data series. After series is converted, we can proceed for further analysis. If this operation is not performed then the series will have increasing pattern with time. We will see adequate increase in the variance, mean in as sample size increases. Thus we can easily underestimate the mean and variance for the future period.
Now for the model number two, we can see how the increasing trend is removed and data becomes stationary. We do not see any visible pattern, trend or cycle by just by observing the data plots. Hence all the data can be said to be not dependent on the location of time interval. Now we have converted the time series to very close to a stationary data series.
We have performed further the test of type augmented_Dickey fuller is performed and this test the prediction is good. We also see that the R_square value has significantly reduced from 0.96 to 0.23, which means that diff_export and diff_import explain only 23% of the variation in diff( GDP ). The intercept indicates the starting amount of ln( GDP ) for the year 1970 which is $e0.06. 0.076 is the standard error of the regression estimate. Thus the model gets improved but still is not fully linear but is more or less linear, thus the time series becomes stationary , and thus will have the same variance, mean, and autocorrelation making it easier to predict the future.
References
1 ) http:// databank . worldbank .org /data / reports.aspx?source= world developmentindicators
2) Faculty.smu.edu,2020
3) https://www.machinelearningplus.com/timeseries/augmenteddickeyfullertest/
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