Quantitative Methods for Finance - Research Paper Example

Quantitative Methods for Finance Introduction This paper discusses a few questions related to the Monte Carlo simulation method. The Monte Carlo method uses techniques like repeated random sampling to generate simulated results for any algorithms or regressions. This method is used extensively in the financial world and depends on a series of randomly generated inputs. This paper discusses three questions pertaining to the analysis of a dynamic regression equation using the Monte Carlo method that aim to evaluate the scenario from many perspectives. Part 1 The first part of the Monte Carlo experiment was conducted using 10, 30 and 100 observations as specified. For better clarity in the results, the number of replications was set to 1500. The bias and the accuracy of the significance tests are provided below for each observation value. a. 10 observations The following is the results obtained for the Ordinary least squares regression performed on the dependent variable ‘y’ for 10 observations. From the above results, the bias for the YSeries is -0.331 and has a very low standard deviation of 0.008. The RMSE (Root Mean Square Error) is an absolute measure of the residuals. A low value indicates a better fit. The RMSE is relatively higher for the Intercept in comparison to the RMSE values for the XSeries and YSeries, suggesting that the latter two parameters exhibit a better absolute fit to the given data. The EDFs from the analysis is shown below: Based on the above results, X5% should be 1.96 for the null hypothesis to be true. However, the upper tail quantiles in the case of both variables (XSeries and YSeries(-1)) lead to the rejection of the null hypothesis since their 5% values exceed the critical value of 1.96. The power of the significance tests in the case of both XSeries and YSeries(-1) is determined by using the following values: The null hypothesis in the case of both the variables is that H0: µ = µ0 for a given value µ0 (sample mean). The alternative hypothesis in either case states that Ha: µ ≠ µ0, indicating a two tailed test. The test statistic is computed using the formula: The power is computed by determining β (the type II error). Power of the test = 1 – β. Power of XSeries: Using H0: µ = 1, Ha: µ = 1.3318, s = 0.897 and n = 10, (β) = p(accept H0 | Ha) = 0.787 The power of XSeries is therefore (1 – β) = 0.213 or 21.3% Power of YSeries(-1): Using H0: µ = 0.5, Ha: µ = 0.169, s = 0.32 and n = 10, (β) = p(accept H0 | Ha) = 0.095 The power of XSeries is therefore (1 – β) = 0.905 or 90.5% The power indicates the probability of rejecting the null hypothesis when the true mean differs from the hypothetical mean. From the above two cases, the power for YSeries(-1) is much greater than that for the XSeries indicating a greater probability for rejection of the null hypothesis in the case of the latter. b. 30 observations The figure below shows the bias for the parameters when configured for 30 observations. It appears that the bias has reduced in the case of both estimators as a result of this increase in the number of observations (compared to the earlier simulation with 10 observations) indicating that the new estimates provide a better fit. The RMSE values have also reduced suggesting a similar conclusion. However, the higher RMSE value for the intercept indicates a comparatively lower fit with respect to XSeries and YSeries(-1) which have much lower RMSE values. Once again, the t-statistics indicate that the 5% estimate exceeds the critical value in the case of both estimators (excluding the intercept). Thus, it appears that the probability of rejecting the null hypothesis (H0) correctly is higher than α. The power of the hypothesis tests is calculated using the following information: Power of XSeries: Using H0: µ = 1, Ha: µ = 1.206, s = 0.488 and n = 30, (β) = p(accept H0 | Ha) = 0.371 The power of XSeries is therefore (1 – β) = 0.629 or 62.9% Power of YSeries(-1): Using H0: µ = 0.5, Ha: µ = 0.394, s = 0.159 and n = 30, (β) = p(accept H0 | Ha) = 0.05 The power of XSeries is therefore (1 – β) = 0.95 or 95% The power of the tests for both parameters has increased suggesting that there is a greater chance of rejecting a null hypothesis when it is false due to an increase in the number of observations. c. 100 observations The bias of the estimators for 100 observations is shown below: It is readily observable that both the bias and RMSE have reduced to very small values (when compared with the values calculated using 30 observations) as a result of an increment in the number of observations. This clearly suggests that the regression assumes a better fit with an increase in the observation size. The t-statistics shown above suggest that the t-statistic values of the estimators exceed the 5% significance level (1.96). The null hypothesis is therefore rejected and the power of the hypothesis tests is calculated below. Power of XSeries: Using H0: µ = 1, Ha: µ = 1.059, s = 0.206 and n = 100, (β) = p(accept H0 | Ha) = 0.183 The power of XSeries is therefore (1 – β) = 0.817 or 81.7% Power of YSeries(-1): Using H0: µ = 0.5, Ha: µ = 0.466, s = 0.084 and n = 100, (β) = p(accept H0 | Ha) = 0.019 The power of XSeries is therefore (1 – β) = 0.981 or 98.1% It is observed that the difference between the true and hypothesized values is much less in comparison to the two earlier cases (10 and 30 observations). As expected, the powers of the tests for both estimators have increased as a result of an increment in the number of observations. Hence it is recommended that the number of observations be maximized to get a regression equation with the best fit. Part 2 This section uses a few common misspecification tests such as the Jarque-Bera (JB) Normality test, the Lagrange Multiplier (LM) test to detect any ARCH effects (known to be sensitive to the presence of an outlier), and the Durbin-Watson (DB) statistic to detect the presence of any first order serial correlation in the residuals. a. In the first case, the value of the omitted regressor y(t-1), λ, is set to 0. The results of the lists mentioned above under this setting are shown below: Jarque-Bera Test, JB = 1.1723 {0.556} Neglected ARCH (LM): F(1,98) = 5.0071 {0.028} Durbin Watson Statistic, d = 0.862521 The Jarque-Bera test is a goodness of fit test that estimates the departure from normality and is calculated from the sample skewness and kurtosis. In the above case, the critical value of the chi-square distribution at the 5% confidence level and with 2 degrees of freedom is 5.99. Since the value of the JB statistic is less than this critical level, the null hypothesis that the residuals are normally distributed is not rejected. The neglected ARCH test is computed from the Engle’s LM test statistic based on the regression of squared residuals on lagged squared residuals. The LM statistic exceeds the critical value of 3.938 thereby rejecting the null hypothesis that no ARCH errors exist. Thus, the alternate hypothesis suggesting the presence of ARCH (Autoregressive Conditional Heteroskedasticity) effects is accepted. Using the DW statistic, one can determine the existence of first order serial correlation, which means that errors pertaining to a given time period are directly correlated to the errors occurring within an ensuing time period. The DW statistic in this case is less than 1. This means that successive error terms are close to each other or positively correlated. b. The value of the omitted regressor y(t-1) = 0.5 (fixed). The results of the various misspecification tests are listed below: Jarque-Bera Test = 0.2978 {0.862} Neglected ARCH (LM): F(1,98) = 1.5611 {0.214} Durbin Watson Statistic = 1.87023 The critical value of the JB test statistic is 5.99. Thus, the null hypothesis that the residuals are not normally distributed is not rejected. The LM test statistic is below the critical value of 3.938. Thus the null hypothesis in this case is not rejected (suggesting that no considerable ARCH errors exist). Changing the value of the omitted variable to 0.5 has also improved the DB test statistic, which is much closer to 2 when compared with the previous case. This indicated the presence of very little autocorrelation. Part 3 The results of the misspecification tests conducted upon including an additional regressor x(-1) are shown below: Jarque-Bera Test = 0.2042 {0.903} Neglected ARCH (LM): F(1,96) = 1.6371 {0.204} Durbin Watson Statistic = 1.96948 Like previous cases, the JB test statistic is less than the critical value (5.99). The null hypothesis that the residuals are not normally distributed is therefore not rejected. The LM test statistic is also less than the critical value (3.94) indicating that there are no considerable ARCH errors. The DW statistic is almost equal 2 suggesting that there is no autocorrelation in the sample. Conclusion The preceding sections provide answers to the various questions specified under qmH4. The first section discusses the bias of the estimator and also the accuracy and significance of the hypothesis tests for different observations. It has generally been observed that the simulation yields better results across all these parameters as the number of observations is increased. The remaining two sections discuss the effects of misspecification associated with the omission or inclusion of a regressor into the equation. These effects have been identified through appropriate techniques such as the Jarque-Bera, Lagrange Multiplier and the Durbin Watson statistic. Read More