CFA II Quantitative Analysis
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Transcript of CFA II Quantitative Analysis
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Qunatitative Analysis – I
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• Regression analysis– Population Regressions Line– Sample Regression Line– Hypothesis Testing– Explained and Unexplained Variation– Residual Analysis
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Agenda
Data
Trend Model
Seasonality
Log-Linear Model
Linear Model
Auto-Regressive
Model
RegressionAnalysis
Cross-sectional Data
Time dependent Data
Significant Autocorrelation among Residuals
Yes
No
Significant Autocorrelation among Residuals
Significant Autocorrelation among lagged
residuals
Yes
Yes
No
No
Make a Graph
Correctly Specified
Model
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Types of Regression Models
Negative Linear Relationship
Negative Linear Relationship
Relationship NOT Linear
No Relationship
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Random Error for this x value
y
x
Observed Value of y for xi
Predicted Value of y for xi
exbby 10
xi
Slope = β1
Intercept = β0
ei
Sample Regression Function
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Sample Regression Function
e x bby 10i
Estimate of the regression intercept
Estimate of the regression slope
Independent variable
Error term
Notice the similarity with the Population Regression Function
Can we do something of the error term?
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• General Multiple Regression Analysis– Hypothesis Testing of Coefficients– Analysis of Variance (ANOVA) and F-statistic– Coefficient of Determination(R2) and Adjusted R2
– Heteroskedasticity, Serial Correlation and Multicollinearity– Model Misspecifications– Models with qualitative dependent variable
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Agenda
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General Multiple Linear Regression Model
• In simple linear regression, the dependent variable was assumed to be dependent on only one variable (independent variable)
• In General Multiple Linear Regression model, the dependent variable derive sits value from two or more than two variable.
• General Multiple Linear Regression model take the following form:
where:
Yi = ith observation of dependent variable Y
Xki = ith observation of kth independent variable X
b0 = intercept term
bk = slope coefficient of kth independent variable
εi = error term of ith observation
n = number of observationsk = total number of independent variables
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ikikiii XbXbXbbY .........22110
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Assumptions of Multiple Regression Model
• There exists a linear relationship between the dependent and independent variables.• The expected value of the error term, conditional on the independent variables is zero.• The error terms are homoskedastic, i.e. the variance of the error terms is constant for all the
observations.• The expected value of the product of error terms is always zero, which implies that the error terms
are uncorrelated with each other.• The error term is normally distributed.• The independent variables doesn’t have any linear relationships between each other.
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Analysis of Variance (ANOVA)
• Analysis of variance is a statistical method for analyzing the variability of the data by breaking the variability into its constituents.
• A typical ANOVA table looks like:
• From the above summary(ANOVA table) we can calculate:
– Standard Error of Estimate(SEE)=
– Coefficient of determination(R2)=
=
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Source of Variability DoF Sum of Squares Mean Sum of SquaresRegression(Explained) 1 RSS MSR=RSS/1
Error(Unexplained) n-2 SSE MSE=SSE/n-2Total n-1 SST=RSS+SSE
2n
SSEMSE
SST)Variation(Total
SSE)Variation(dUnexplaineSST)Variation(Total
SST)Variation(Total
RSS)Variation( Explained
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• Time Series Analysis– Trend Models– Autoregressive Models– Seasonality– Random Walk Process
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Agenda
Data
Trend Model
Seasonality
Log-Linear Model
Linear Model
Auto-Regressive
Model
RegressionAnalysis
Cross-sectional Data
Time dependent Data
Significant Autocorrelation among Residuals
Yes
No
Significant Autocorrelation among Residuals
Significant Autocorrelation among lagged
residuals
Yes
Yes
No
No
Make a Graph
Correctly Specified
Model
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Time Series
• Time Series is a series of the variable values taken at equal interval of time. The closing price of the IBM stock observed for 10 years constitutes the time series of the IBM stock price.
• Time series may have a pattern when plotted against the time, which depicts the characteristics of the IBM stock and the capital markets in general.
• These patterns in the stock price time series are called trends in the time series,– An American retail chain (AMR) which sells woolen clothes, will have increased sales pattern in the
winters and a moderate sales in the summer.
– The quarterly sales of AMR when plotted against the time for 4 years will show moderate sales in summers and increased sales in winters
• The above shown trends in the sales are called seasonal trends
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QuarterlySales
Time 4th Quarter
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Limitation of Trend Models
• The most important assumption of the linear regression is that the error terms are not correlated with each other.
• Other important assumption of the linear regression is that the residual term is independently distributed.
• These two important assumptions when violated becomes the limitation for the trend models as linear regression is used in trend models.
• To overcome the autocorrelation problem(violation of independent and uncorrelated residuals assumption), log-linear trend model can be used which reduces the serial correlation.
• After applying the log-linear trend model, the serial correlation may persist, which means even a log-linear trend model is inappropriate for the case. This hints us to use some other form model which are autoregressive models.
• In Autoregressive(AR) models, the dependent variable is regressed with its lagged term.
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