CFA Level 2 – Quantitative Analysis
Testing the Existence of a Significant Relationship between
Dependent Variable and Independent Variable
1.
Test on Population Slope 
=0
≠0

H0:
H1:

Test Statistic:
(There is no relationship)
(There is a relationship)
t
b1  β1
sb1
t
b1
sb1
where Sb1 is the Sample Standard Deviation for Sample Slope

2.
If the t-ratio > the critical t-statistics, then H0 is rejected (Beta is significantly different
from zero) and evidence of a linear relationship is concluded.
Test on the Existence of a Significant Correlation, 
=0
≠0

H0:
H1:
(There is no relationship)
(There is a relationship)

Test Statistic:
t
ρ n2
1  ρ2
where (n – 2) = Degrees of Freedom

3.
If the t-ratio > the critical t-statistics, then H0 is rejected (Correlation Coefficient is
significantly different from zero) and evidence of a linear relationship is concluded.
Test on Confidence-Interval of 
=0
≠0

H0:
H1:
(There is no relationship)
(There is a relationship)

Test Statistic:

If the b1  tn-2Sb1 > Zero, then H0 is rejected (Beta is significantly different from zero) and
evidence of a linear relationship is concluded.
b1  tn-2Sb1
P-VALUE APPROACH


The p-value approach involve calculating the value of the specified test statistic and the
corresponding p-value from sample data, then comparing with the chosen significance level .
If the p-value < than the significant level , reject the H0.
F-STATISTIC

F-Statistic – The F-statistic is used to test the hypothesis that all the regression coefficients
of the independent variables are simultaneously equal to zero. Expressed mathematically,
the F-statistic is used to test the null hypothesis that H0: Beta1=Beta2=Beta3=0. (i.e. all the slope
coefficients in the regression equation are significant as a group)

For Multiple Regression Model

If the F-statistic > the critical value, then H0 is rejected
1
Norman Cheung
CFA Level 2 – Quantitative Analysis
T-STATISTIC


T-Statistic – The T-statistic is used to test the hypothesis that the only one regression
coefficient of a Linear Model or one single of a Multiple Regression Model is equal to zero
(Note the difference bet. T-test & F-test). Expressed mathematically, the T-statistic is used to
test the null hypothesis that H0: Beta1=0.
If the T-statistic > the critical value, then H0 is rejected
Interpretation of R2 – “Goodness of Fit” or Coefficient of Determination

The R2 indicates the percentage of the total variability of the dependent variable that can be
explained by the regression model.
INTERCEPT AND SLOPE COEFFICIENT


The intercept coefficient may be interpreted as the level of the dependent variable given a
zero value for the independent variable. This interpretation assumes that the prediction occurs
within the range of values of the independent variable used in developing the regression
equation.
The slope coefficient represents the change in the dependent variable given a unit change in
independent variable.
AUTOCORRELATION

Autocorrelation occurs when successive observations of the dependent variable are correlated
and tends to occur when data are collected over a period of time, and a residual analysis is
required to determine if autocorrelation exists.
MULTICOLLINEARITY



Multicollinearity exists when there is a high degree of correlation among the independent
variables in a regression model.
Symptoms of high multicollinearity include:
1. Large R2 but statistically insignificant regression coefficients.
2. Regression coefficients that change greatly in value when independent variables are
dropped or added to the equation
3. The magnitude of one or more coefficients is unexpectedly large or small relative to
expectations.
4. A coefficient appears to have a “wrong” sign (the sign is counterintuitive).
The Danger of Multicollinearity – is that the coefficients of the regression equation may be
unstable. When multicollinearity exists in a model, it is difficult to determine which
independent variables are contributing significantly to the explanatory power of the model.
Therefore, the reliability of this model for explaining which variables contribute to the forecast
of the value of the dependent variable.
STATISTICALLY SIGNIFICANT AND ECONOMICALLY SIGNIFICANT


Statistically Significant – do have a relationship between the dependent and independent
variables, say a linear relationship.
Economically Significant – the impact of the independent variable on the dependent is
significant in terms of quantity; i.e., the regression coefficient (or estimate of the dependent
variable) is large enough, say 100, 200 instead of 0.0001 or 0.0002 etc .
2
Norman Cheung
CFA Level 2 – Quantitative Analysis
ANOVA – ANALYSIS OF VARIANCE TABLE
Source of Variation
Regression (R)
Sampling Error (E)
Total (T)
Degrees of Freedom
(df)
Sum of Squares
k
SSR
n - (k+1)
SSE
n-1
SST
Mean Square
MSE 
MSE 
F-Statistics
SSR
k
F
MSR
MSE
SSE
n  ( k  1)
Standard Error of the Regression Model
 MSE 
SSE
n  ( k  1)
More CFA info & materials can be retrieved from the followings:
For visitors from Hong Kong: http://normancafe.uhome.net/StudyRoom.htm
For visitors outside Hong Kong: http://www.angelfire.com/nc3/normancafe/StudyRoom.htm
3
Norman Cheung