KENYATTA UNIVERSITY
DEPARTMENT OF ECONOMETRICS AND STATISTICS
EES 801 ECONOMETRICS II – ASSIGNMENT 2
Name of the Student
Student ID
Date
FOR ASSISTANCE: glmwriters@gmail.com
QUESTION ONE
a). Figure 1 below shows the time series graph for Dow Jones, Nasdaq and S&P 500 from 1990q1 to 2023q4. The findings indicate that, in general, the stock prices increased over time. However, the general increase was higher for Dow Jones compared to the Nasdaq and S&P 500.
Figure 1
US Stock Prices over Time
b). The stock prices exhibit non-stationary behaviour. Worden et al. (2019) observe that the stationarity can be observed using the Augmented Dickey-Fuller test (ADF). The test was computed using STATA, and the output is attached (Appendix A). Based on the output, the p-values were;
Dow Jones – p-value = 0.9934
Nasdaq – p-value = 0.9666
S&P 500 – p-value = 0.9969
The p values for each of the variables were high, above 0.01 and 0.05, which means that the null hypothesis of no non-stationarity is rejected, implying that the time series is non-stationary.
c). The test of co-integration was achieved through regression by using the S&P 500 as the left-hand or dependent variable. The results of the regression analysis are described in Appendix B below. Residuals from the regression analysis were subjected to the ADF test. The test indicates critical values at the 1%, 5% and 10% levels of significance. The p-value was high at p = 0.5181, which implies that there was no co-integration at each of the levels.
d). The test applied was the Engle-Granger Two-Step Co-integration Test. The test is considered appropriate in situations where the dependent variable is clearly defined (Schaffer, 2022). In the current study, the left-hand variable was clearly defined, which justifies the use of the Engle-Granger test.
e). On the assumption that co-integration exists, the ARDL model was selected. Considering S&P 500 as the dependent variable, the model selected by STATA was ARDL (1,1,1). The ARDL (1,1,1) model was used to estimate the error correction term model. As evident in Appendix C below, the coefficient value was -0.085 with p-value less than 0.05 implying that any deviations in the data in the long-term may have been corrected in the short-term.
QUESTION TWO
a). Figure 2 below shows the time series plot of the fed rate against the bond rate in the US stock prices. The findings in Figure 2 illustrate that the two rates moved together over the period of focus. This means that over this period, the bond rate for US stock prices changed with the changing Fed rates. In this case, the fed rates influence economic conditions and inflation, which in turn influences how investors react to bonds.
b). A dataset is considered to be stationary when the main statistical aspects such as the mean and variance remains the same and do not change over time. It means that the statistical properties do not change over time (Worden et al. 2019). On the contrary, a dataset is considered to be non-stationary when the statistical properties such as mean and variance tend to change over time. The lack of stationarity makes it difficult to predict patterns in the datasets.
C). The test for stationarity was achieved using the ADF test in STATA. Appendix D below shows the results from ADF where in both variables (fed rate, p = 0.1073) and (bond rate, p = 0.0502), the p-values were above 0.05 which confirms non-stationarity at 5% critical level. The two series would therefore be considered to be non-stationary.
- d) The ADF tests above have confirmed that the datasets have a unit root for (p > 0.05). Considering the ADF test formula
ΔYt = α + γYt–1 + ∑δiΔYt−I + εt
The unit root is confirmed in situations where γ = 0
- e) The test of co-integration was achieved through regression by using bond rate as the left-hand or dependent variable. The results of the regression analysis are described in Appendix E below. Residuals from the regression analysis were subjected to the ADF test. The test indicates critical values at the 1%, 5% and 10% levels of significance. The p-value was high at p = 0.0001, which implies that there is integration at each of the levels. The findings imply that while initially the time series dataset was found to be non-stationary, co-integration exists between the variables, which is an indication of a long-term relationship between variables.
QUESTION THREE
a). The binary probit model applied in the question could be defined as;
Y = Xi × β + ϵi
Where Y is the dependent variable, represented by 1 for currency crush and zero otherwise
X represents the independent variables (debt composition, domestic macroeconomics, external and foreign variables)
β represents the vector of coefficients
ε represents the error term
b). The probability that a country would experience a currency crash would be determined by considering
P(Yi = 1) = P(Xi × β + ϵi > 0)
For Yi = 1
Assuming that both the dependent variable and the error term follow a normal distribution
c). The loglikelihood for the model is described as
ln L(β) = ∑i=1N [Yi ln Φ (Xiβ) + (1−Yi) ln (1−Φ (Xiβ))]
In this case, the maximum likelihood ML function is obtained by maximizing the function
d). The findings in the debt composition variables are an indication of how a one percentage change affects the overall likelihood of a currency crash, holding all other variables constant. The findings would be interpreted by considering the z values. At the 95% level, the focus is on whether the values are below or above 1.96. Out of the seven debt composition variables, only one, FDI/Debt, was found to have a z value above 1.96, indicating that its impact was significant. For this variable, the probability is -0.33, which means that a 1% increase in the FDI/Debt would reduce the probability of currency crash by 0.33%. All other variables were found to have an insignificant effect for z less than 1.96. On probabilities, the values for the commercial bank, concessional, FDI/debt and multilateral debts were negative, indicating that they would reduce the probability of a currency crash. On the contrary, the values for variable rate, short-term and public sector debts are positive, indicating that they would increase the probability of a currency crash.
e). From the study, the factors that are likely to trigger a currency crash are statistically significant. The factors include FDI/Debt, international reserve imports, real output per capita growth, foreign interest rate and domestic currency growth. In this case, a currency crash is likely to be experienced during periods of low FDI to debt, low international reserves to imports and low real output per capita growth, as these variables have negative probabilities. At the same time, according to the study, a currency crash is likely to be experienced during periods of high growth in domestic credit and foreign interest rates.
References
Schaffer, M. (2022). egranger: Stata module to perform Engle-Granger co-integration tests and 2-step ECM estimation.
Worden, K., Iakovidis, I., & Cross, E. J. (2019). On stationarity and the interpretation of the ADF statistic. In Dynamics of Civil Structures, Volume 2: Proceedings of the 36th IMAC, A Conference and Exposition on Structural Dynamics 2018 (pp. 29-38). Springer International Publishing.
Appendixes
Appendix A: ADF Test Question 1
Appendix B: Co-integration
Appendix C: ARDL-ECM Model
Appendix D: ADF Question Two