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 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


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


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


• 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


Yes

No


Significant Autocorrelation among lagged

residuals

Yes

Yes

No

No

Make a Graph

Correctly Specified

Model


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


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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cfa ii quantitative analysis

Education

loglinear trend model

simple linear regression

linear relationships

specified model

time series time series

variable independent

population regression

thindependent variable