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L2 flash cards quantitative methods - SS3

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  • 1. Scatter Plot scatter plot - a graphical representation of the relationship between the two variables Study Session 3, Reading 11
  • 2. Correlation and Covariance Analysis Correlation analysis - expresses the relationship between two variables with the help of a single number. It measures both the extent and direction of the linear relationship between two variables. Formula: Sample covariance of X and Y for a sample size of ‘n’ can be calculated as: Study Session 3, Reading 11
  • 3. Correlation and Covariance Analysis (cont.) Formula: Sample Correlation Coefficient: Where: SX - standard deviation of variable X SY - standard deviation of variable Y Formula: Sample Standard Deviation Study Session 3, Reading 11
  • 4. Limitations to Correlation Analysis Outliners Outliers are a small number of observations that are at either extreme of a sample Spurious Correlation  The correlation between two variables that shows a chance relationship in a particular data set is called spurious correlation.  The correlation between two variables that arises not from a direct relationship between them but their relation to third variable is also called spurious correlation. Study Session 3, Reading 11
  • 5. Hypothesis Testing For Population Correlation Coefficient Proposed Hypothesis:  null hypothesis - H0 , that the correlation is 0 (p=0)  alternative hypothesis - Ha that the correlation of population is different from 0 (p≠0) Formula: t-test Study Session 3, Reading 11
  • 6. Dependent and Independent Variables in Linear Regression independent variable (denoted as X) - the variable that is used to explain changes dependent variable (denoted as Y) - the variable that is to be explained. Linear regression involves the use of one variable to make a prediction about other variable. It also involves testing hypotheses about the relation between the two variables and quantifying the strength of relationship between the two variables. Study Session 3, Reading 11
  • 7. Dependent and Independent Variables in Linear Regression (cont.) Regression equation that defines the linear relation between the dependant and independent variable: Where: Y - dependent variable b0 – intercept b1 - slope coefficient X - independent variable - error term Study Session 3, Reading 11
  • 8. Dependent and Independent Variables in Linear Regression (cont.) In linear regression, estimated or fitted parameters b0 and b1 are chosen in the given equation to minimize: cross sectional data - uses many observations on the dependant and independent variables for the same time period time-series data - many observations from different time periods are used Study Session 3, Reading 11
  • 9. Assumptions of a Classical Linear Regression Model 1. There is a linear relationship between the independent and 2. 3. 4. 5. 6. dependant variable. The independent variable is not random. The expected value of the error term is 0. The error term is normally distributed. The error term is uncorrelated across observations. The variance of the error term is the same for all observations (Homoskedasticity Assumption). Study Session 3, Reading 11
  • 10. Standard Error of Estimate Standard Error of Estimate (also called the standard error of regression) - used to measure how accurately a regression model fits the data. Formula: Study Session 3, Reading 11
  • 11. Coefficient of Determination coefficient of determination - used in measuring the proportion variance in the dependent variable that is explained by the independent variable Formula: Study Session 3, Reading 11
  • 12. Confidence Interval for Regression Coefficient regression coefficient - the average change in the dependant variable for every one unit change in the value of the independent variable. Things needed to estimate confidence interval for the regression coefficient:  estimated parameter value for a sample  standard error of estimate  Significance level for t-distribution  degree of freedom (n-2). Formula: Where: tc - critical t value at a chosen significant level Study Session 3, Reading 11
  • 13. Hypothesis Testing for a Population Value of the Regression Coefficient Formula: When testing a hypothesis using a regression model with t-test of significance, the t statistic is computed as: Formula: The confidence interval for the test is given as: Study Session 3, Reading 11
  • 14. Calculating a Predicted Value for the Dependent Variable Two sources of uncertainty in using regression model: 1. the error term 2. estimated parameters ( bˆ0 and bˆ1 ) Given the regression model Yi =bo +b1 Xi +Ei , if estimated parameters bˆ0 and bˆ1 are known, the predicted value of dependent variable ,Y, can be calculated as: Study Session 3, Reading 11
  • 15. Calculating a Predicted Value for the Dependent Variable (cont.) The prediction interval for a regression equation for a particular predicted value of the dependent variable is computed as: Where: Sf - square root of estimated variance of prediction error tc - critical level for t-statistic at chosen significance level The confidence level is taken as Study Session 3, Reading 11
  • 16. Calculating a Predicted Value for the Dependent Variable (cont.) The estimated variance of the prediction error ( calculated as: of Y) is Where: S2 - squared standard error of estimate -variance of independent variable Study Session 3, Reading 11
  • 17. Calculating ANOVA in Regression Analysis Analysis of Variance (ANOVA) - a statistical procedure that is used to determine how well the independent variable or variables explain the variation in the dependant variable. F-test - the statistical test that is used in the analysis of the variance Study Session 3, Reading 11
  • 18. F-test A F-statistic is used to test whether the slope coefficients in a linear regression are equal to 0 or not. In a regression equation with one independent variable:  Null Hypothesis H0 : b1= 0  Alternative Hypothesis Ha : b1≠ 0 Things required to undertake an F-test 1. the total number of observations 2. the total number of parameters to be estimated 3. the sum of squared errors(SSE) 4. regression sum of squares (RSS) Study Session 3, Reading 11
  • 19. F-test (cont.) Formula: SSE Formula: RSS Formula: Total Variation (TSS) = SSE + RSS Formula: F-statistic in a regression with one independent variable Study Session 3, Reading 11
  • 20. Limitations to Regression Analysis Parameter instability In investment analysis, regression models can have limited use because public knowledge of regression relationships can negate their use for future purpose Violations of assumptions can make hypothesis tests and predictions invalid Study Session 3, Reading 11
  • 21. Multiple Regression Equation multiple regression equation - used to determine how a dependent variable is affected by more than one independent variables log-log regression model - used when the proportional changes in the dependent variable bear a constant relationship to a proportional changes in independent variables General Form of the Multiple Regression Model Where: Yi - the ith observation of the dependent variable Y Xji - the ith observation of the independent variable Xj, j=1,2,…,k b0 - the intercept of the equation b1 ,…., bk - the slope coefficients for each of the independent variables Ei - the error term n - the number of observations Study Session 3, Reading 112
  • 22. Hypothesis Testing for a Population Value of a Regression Coefficient Under the null hypothesis, the hypothesis population value of a regression coefficient is taken as 0. The degrees of freedom in the test are the number of observations minus the number of independent variables + 1 (i.e. n – (k+1).) Formula: Hypothesis testing using t-test: Where: b^j - regression estimate of hypothesized value of coefficient -estimated standard error of b^j Study Session 3, Reading 12
  • 23. Hypothesis Testing for a Population Value of a Regression Coefficient (cont.) p-value The p-value for a regression coefficient is the smallest level of significance at which the null hypothesis of that population value of the coefficient is 0 can be rejected in a two-sided test. The lower the p-level, the more accurate the result of the test. Study Session 3, Reading 12
  • 24. Confidence Interval for the Population Value and Predicted Value for the Dependent Variable Two types of uncertainty in predicting the dependent variable using linear regression model:  the regression model itself because of standard error of estimate  uncertainty about estimates of regression model parameters The computation of the prediction interval to accommodate the uncertainties is done with the help of matrix algebra. Study Session 3, Reading 12
  • 25. Points to be considered for predicting a dependent variable Assumptions required for using a regression model must be met. Caution should be exercised on predictions that are based on the value of independent variables that are outside the range of data used for estimating the model. Study Session 3, Reading 12
  • 26. Steps in predicting the value of the dependent variable Obtaining estimates of regression parameters ( ). Determining assumed values of independent variables Computing predicted value of dependent variable using the equation: Study Session 3, Reading 12
  • 27. Assumptions of a Multiple Regression Model 1. There exists a linear relationship between the dependent variable 2. 3. 4. 5. 6. and the independent variables. There is no exact linear relationship between two or more of the independent variables and the independent variables are not random. The error term is normally distributed. The error term is uncorrelated across observations. The variance of the error term is the same for all of the observations. The expected value of error term, conditioned upon the independent variable, is 0. Study Session 3, Reading 12
  • 28. F-statistic in Regression Analysis F-statistic - used to test whether at least one of the slope coefficients of the independent variables is not equal to 0 null hypothesis - all the slope coefficients in the multiple regression model are equal to 0 is presented as : alternative hypothesis - at least one slope coefficient is not equal to 0. Study Session 3, Reading 12
  • 29. F-statistic in Regression Analysis (cont.) Things required for F-test Total number of observations (n). Total number of regression coefficients to be estimated (k+1) where k is number of slope coefficients. Sum of squared errors (SSE) (Unexplained Variation) Regression sum of squares (RSS) (Explained Variation) Study Session 3, Reading 12
  • 30. F-statistic in Regression Analysis (cont.) Calculating the F-statistic Degrees of freedom in the test 1) k (numerator degrees of freedom) 2) n-(k+1) (denominator degrees of freedom) Study Session 3, Reading 12
  • 31. R2 and Adjusted R2 in Multiple Regression R2 measures how appropriately the regression model fits with one independent variable. Adjusted R2 ( ) is used in place of R2 when there is more than independent variable. Relationship: Where: n - the number of observations k - number of independent variables Study Session 3, Reading 12
  • 32. Dummy Variables Dummy variables - used in regression models to determine whether a qualitative independent variable explains the dependent variable A dummy variable has a value of 1 if a particular qualitative condition is true and 0 if that condition is false. In order to distinguish between n categories, n – 1 dummy variables are required. Study Session 3, Reading 12
  • 33. Heteroskedasticity and its Effect on Statistical Inference Heteroskedasticity - a violation of the regression assumption that the variance of the errors in a regression is constant across observations. Two types of heteroskedasticity : 1. unconditional heteroskedasticity 2. conditional heteroskedasticity Breusch-Pagan test - widely used when testing for conditional heteroskedasticity. Two methods used for correcting conditional heteroskedasticity: 1. Robust Standard Errors 2. Generalized Least Squares Study Session 3, Reading 12
  • 34. Heteroskedasticity and its Effect on Statistical Inference (cont.) Durbin-Waston test – test conducted when serial correlation generally arises in time-series regressions Consequences of Heteroskedasticity F-test does not provide reliable results. T-tests for the significance of individual regression coefficients does not provide reliable results. Standard errors and test statistics will have to be adjusted in order to derive reliable results. Study Session 3, Reading 12
  • 35. Unconditional Heteroskedasticity and Conditional Heteroskedasticity Unconditional heteroskedasticity arises when the heteroskedasticity of an error variance does not correlate with the independent variables. This heteroskedasticity is not a major problem for statistical inference. Conditional heteroskedasticity arises when heteroskedasticity in the error variance is correlated with the independent variables. This heteroskedasticity is a major problem for statistical inference. Study Session 3, Reading 12
  • 36. Methods for Correcting for Heteroskedasticity 1. Under the robust standard error method, the standard errors of a linear regression model’s estimated coefficients are corrected. 2. Under the generalized least square method, original equation is modified and a new modified regression equation is estimated. Study Session 3, Reading 12
  • 37. Consequences of Serial Correlation Incorrect estimates of the regression coefficient standard errors. If the independent variable is a lagged value of the dependent variable, it will make the parameter estimates invalid. In positive serial correlation, a positive (negative) error for one observation increases the positive (negative) error for another observation. Positive serial correlation has no effect on the consistency of estimated regression coefficients, but affects validity of statistical tests. Study Session 3, Reading 12
  • 38. Durbin-Waston Test Formula: Study Session 3, Reading 12
  • 39. Methods to correct for Serial Correlation 1. The coefficient standard errors for the linear regression parameter estimates can be adjusted. 2. Regression equation can be modified to eliminate serial correlation. Study Session 3, Reading 12
  • 40. Multicollinearity in Regression Analysis Multicollinearity - a violation of the regression assumption that there is no exact linear relationship between two or more independent variables Consequences of Multicollinearity Estimates of regression coefficients become unreliable. It is not possible to ascertain how individual independent variables affect dependent variables. Study Session 3, Reading 12
  • 41. Model Misspecification in Regression Analysis Model specification - the set of variables that are included in the regression and the regression equation’s functional form Misspecified Functional Form It omits one or more important variables from regression. One or more regression variables are required to be transformed before estimating the regression. Data has been pooled from different samples that are not to be pooled. Study Session 3, Reading 12
  • 42. Model Misspecification in Regression Analysis (cont.) Reasons for time-series misspecification Inclusion of lagged dependent variables as independent variables in regressions which have serially correlated errors. The dependent variable being included as an independent variable. If there are independent variables that are measured with errors. Study Session 3, Reading 12
  • 43. Models With Qualitative Dependent Variables Qualitative dependent variables are dummy variables that are used as dependent variables. 1. Probit model - used to estimate the probability of a discrete outcome when values of independent variables used to explain the outcomes given based on normal distribution 2. Logic model - used to estimate the probability of a discrete outcome when values of independent variables used to explain the outcomes given based on logical distribution Study Session 3, Reading 12
  • 44. Calculating the Predicted Trend Value for a Time Series Linear Trend Models - the dependent variable changes at a constant rate with time Formula: Where: yt - value of the time series at time t b0 - the y-intercept term b1 - the slope coefficient (trend coefficient) t - time (independent variable) Et - a random error term Study Session 3, Reading 13
  • 45. Calculating the Predicted Trend Value for a Time Series (cont.) Log-Linear Trend Models - used when the time series tends to grow at a constant rate Formula: Predicted trend value of Study Session 3, Reading 13
  • 46. Limitations of the Use of Trend Models for a Given Time Series Trend models can suffer from the limitation of serially correlated errors. If trend models have errors that are serially correlated, better forecast models for such time series are required than trend models. Study Session 3, Reading 13
  • 47. Covariance Stationary Following things should be finite and constant in all periods:  Expected value of time series.  Variance of time series.  Covariance of time series with itself for a fixed number of periods in the past or future. Implications if the Time Series is not Covariance Stationary  Estimate of autoregressive time series by using linear regression will not be valid  The hypothesis test will provide invalid results. Study Session 3, Reading 13
  • 48. Structure of an Autoregressive Model of Order p In an autoregressive model, a time series is regressed on its past values and shows the relationship between current period-values and past-period values. pth-order Autoregressive Model: First Order Autoregression Study Session 3, Reading 13
  • 49. Autocorrelation for Time Series Autocorrelation of a time series - the correlation of the time series with its past values Formula: Study Session 3, Reading 13
  • 50. Autocorrelation for Error Term Error autocorrelation is estimated by using sample autocorrelations of the residuals called residual autocorrelations and their sample variance. Formula: Study Session 3, Reading 13
  • 51. Mean Reversion A time series shows mean reversion if it tends to rise when its level is below its mean and falls when its level is above its mean. Formula: Mean Reverting Level Study Session 3, Reading 13
  • 52. Mean Reversion (cont.) Interpretation of Mean Reversion Level If the current value of time series is b0 /(1 – b1 ) , it will neither increase nor decrease. If the current value is below b0 /(1 – b1 ) , the time series will increase. If the current value is above b0 /(1 – b1 ), the time series will decrease. Study Session 3, Reading 13
  • 53. Mean Reversion (cont.) Multiple Periods of Forecasting and the Chain Rule of Forecasting Formula: AR Model Formula: Two-period ahead forecast Study Session 3, Reading 13
  • 54. In-Sample and Out-of-Sample Forecasts In-sample forecasts can be defined as the in-sample predicted values from the estimated time series model. Out-of-sample forecasts are made from estimated time-series models for a period that is different from the period from which the model was estimated. Root Mean Squared Error (RMSE) (calculated as square root of average squared error) - used for comparing the out-ofsample forecasting accuracy of different time series models. Study Session 3, Reading 13
  • 55. Instability of Coefficients in Time-Series Models Generally unstable across different sample periods Different between models that are estimated based on longer or shorter sample periods Depends upon the sample period Study Session 3, Reading 13
  • 56. Random Walk random walk - a time series model in which the value of a series in one period is calculated as the value of the series in the previous period plus an unpredictable random error Formula: Random walk with a drift increases or decreases by a constant amount in each period Formula: Study Session 3, Reading 13
  • 57. Random Walk (cont.) First-differencing - differencing a time series by creating a new time series that in each period is equal to the difference between xt and xt-1. Formula: Study Session 3, Reading 13
  • 58. Dickey Fuller Unit Root Test Formula: Where: g1 = (b1 – 1) Null Hypothesis is H0 : g1 = 0 Alternative Hypothesis is Ha : g1 < 0 Study Session 3, Reading 13
  • 59. Seasonality in a Time-Series Model Seasonality of time series occurs when regular patterns of movement within the year are observed. Formula: Seasonal lag in autoregressive model Formula: Forecasted Value Study Session 3, Reading 13
  • 60. ARCH Models Autoregressive Conditional Heteroskedasticity (ARCH) - if the variance of errors in a time series model depends on the variance of previous Formula: Linear regression error Where: u1 = error term Study Session 3, Reading 13
  • 61. ARCH Models (cont.) Predicting Variance of Errors Formula: Formula: Calculate the variance of the error term in the current period Study Session 3, Reading 13
  • 62. Analysis of Time-Series Variables Prior To Linear Regression Two time series - said to be cointegrated if there is such a longterm financial or economic relationship between the two variables that they do not diverge from each other without being bound in the long run. The (Engle Granger) Dickey Fuller test is used to determine whether time series are cointegrated. Study Session 3, Reading 13
  • 63. Analysis of the Appropriate Time-Series Model Given an Investment Problem Regression models or time series models can be used in the analysis of investment problems. In a regression model, predicting the future value of a variable is undertaken on the basis of a hypothesized casual relationship with other variables. In time series mode, the future behavior of the variable is made on the basis of past behavior of that variable. Study Session 3, Reading 13
  • 64. Explanation of the Dependent Variable by Analysing the Regression Equation and ANOVA Table Key analysis of variance (ANOVA) - used to provide information about a regression model’s explanatory power F-statics are used to test the explanatory power of the dependent variable If independent variables do not explain the dependent variables, the value of the F-statistic is 0. Variability in values of the dependent variable can be divided into two parts: Total Sum of Squares = Regression Sum of Squares + Residual Sum of Squares Study Session 3, Reading 13
  • 65. Uses of Multiple Regression Analysis in Financial Analysis Used in various finance and investment decisions The effect of various parameters on investment decisions can be measured To predict the expected return of a fund or portfolio Dummy variable can be used in various financial analysis models If there are any violations of assumptions, they should be adjusted by analysts before making any decisions Study Session 3, Reading 13