It can be noted from the above results, both methods will cause the function ln(S(t))/t will converge to the limit which is approximately 0.0558077, which is close to the true value of p of 0.055. The true value can be found based on the property of the probability distribution of ln(S(t)). We note that: as S(0)=1, , and Using the Delta Method, we can deduce the distribution of ln(S(t))/t as: Taking limits as t goes towards infinity, we note that: So as t goes to infinity, the variance will reduce to 0, meaning ln(S(t))/t will approach the mean which is 0.055. Thus the true value of p is 0.055 and verified the results of the simulation.
b) To find the value of t to obtain 2 digit accuracy with 95% probability, we find t such that: This is equivalent to finding: Rearranging gives Noting that from part a) that , where p=v, we can deduce that . This implies that , and noting that and , we deduce that . So t must be around 4571822 to obtain two-digit accuracy of the true value of p with approximately 95% probability. c) Based on the values of S(t) simulated from both methods, the value of can also be simulated using both methods. The code can be found in Appendix 2.1.3.
It can be observed from the above graphs that the function does not converge to a limit. The function will continually fluctuate randomly, following a distribution similar to Gamma random distribution. The behaviour can be verified by analysing the properties of probability distribution of , which we obtain from part (a) as: This implies that. It can then be shown that here denotes the Gamma distribution with parameters 0.5 and 0.18. The full derivation can be found in the Appendix 2.1.2. Clearly the behaviour of is independent of t, and thus the function does not approach a limit as t tends to infinity.
2. Weekly exchange rate analysis - US Dollar and pound sterling (1980-1988) Before constructing a stochastic model to fit the data, the data is plotted as a times series, and the sample autocorrelation function (ACF) and sample partial autocorrelation function (PACF) are plotted to see if there is any observable trend or seasonality. It can be noted that from the above plot of the time series of weekly exchange rates, there does not seem to be any significant trend or seasonality. From the sample ACF plot, the correlations decrease rather slowly and steadily from 1, indicating the exchange rates are not stationary. Thus differencing the data may be advisable before fitting a model. The time series of the differenced data is then plotted, as well as the sample ACF and sample PACF plots.
The above plots indicate the differenced data for exchange rates are stationary, as the ACF rapidly goes towards zero after lag 1, while the partial ACF is within the , with This suggests an ARIMA(0,1,0) model may be an appropriate model for exchange rates.When the ARIMA(0,1,0) was fitted, there are certain criteria that needs to be analysed. The AIC (Akaike Information Criteria) for this model is -2035.93 (Other information can be found in the Appendix). To see that this model minimises the AIC, the AIC from the ARIMA(0,1,1) and ARIMA(1,1,0) models are found and compared with the AIC for the proposed model. The AIC from ARIMA(0,1,1) model is -2035.74 while the AIC from ARIMA(1,1,0) model is -2035.79. The AIC of the proposed ARIMA(0,1,0) model is clearly smaller than the AIC of the other 2 models.
Another way to assess the model is to assess the behaviour of the residuals from the fitted model. The standardised residuals, sample ACF of residuals and p values of Ljung-Box (Portmanteau) statistics are plotted on the graph below. From the residual plot, it can be observed that the residuals form a good approximation to a white noise process, with no real seasonal pattern in fluctuation or any trend in the residuals.
The ACF of Residuals plot shows that most of the sample autocorrelations lie within the 95% confidence interval indicated by the dotted lines. The p values for the Ljung-Box (Portmanteau) statistics are all outside the rejection region, which means that the null hypothesis, which states the residuals are white noise with zero mean and variance 1, is accepted. The three above plots all indicate that ARIMA(0,1,0) is an appropriate model of the time series of exchange rates.
Based on the analysis of the residuals for the fitted model and the AIC, it can be concluded that the exchange rates can be fitted using an ARIMA(0,1,0) model, which implies , where is the exchange rate at time t, B is the backshift operator and are uncorrelated white noise. This is equivalent to, which suggests that exchange rates follow a random walk process, which is typical for most financial data. Thus, most economists will try to simulate exchange rate movements using the random walk (ARIMA(0,1,0)) process. This analysis was performed in R. The code and further results can be found in the Appendix.