CFA二级培训项目
Quantitative Methods
CFA二级课程框架
10
Study Session 1-2
Ethics & Professional Standards
Study Session 3
Quantitative Analysis
5-10
Study Session 4
Economics
5-10
Study Session 5-7
Financial Reporting and Analysis
15-25
Study Session 8-9
Corporate Finance
5-15
Study Session 18
Portfolio Management and Wealth Planning
5-15
Study Session 10-12
Equity Investment
20-30
Study Session 14-15
Fixed Income
5-15
Study Session 16-17
Derivatives
5-15
Study Session 13
Alternative Investments
5-15
Total: 100
2-79
Summary of Readings and Framework
SS 3
R9 Correlation and regression
R10 Multiple regression and issues in regression analysis
R11 Time-series analysis
 R12 Excerpt from ’’Probabilistic Approaches: Scenario
Analysis, Decision Trees, and Simulation’’
3-79
Framework
 Correlation and Regression
1. Scatter Plots
2. Covariance and Correlation
3. Interpretations of Correlation Coefficients
4. Significance Test of the Correlation
5. Limitations to Correlation Analysis
6. The Basics of Simple Linear Regression
7. Interpretation of regression coefficients
8. Standard Error of Estimate & Coefficient of Determination (R2)
9. Analysis of Variance (ANOVA)
10. Regression coefficient confidence interval
11. Hypothesis Testing about the Regression Coefficient
12. Predicted Value of the Dependent Variable
13. Limitations of Regression Analysis
4-79
Scatter Plots
 A scatter plots is a graph that shows the relationship between the
observations for two data series in two dimensions.
5-79
Covariance and Correlation
 Covariance:

Covariance measures how one random variable moves with another
random variable. ----It captures the linear relationship.
n


Cov( X , Y )   ( X i  X )(Yi  Y ) /( n  1)
i 1
Covariance ranges from negative infinity to positive infinity
 Correlation:
Cov( X , Y )
r
sx s y

Correlation measures the linear relationship between two random
variables

Correlation has no units, ranges from –1 to +1
6-79
Interpretations of Correlation Coefficients
 The correlation coefficient is a measure of linear association.
 It is a simple number with no unit of measurement attached, so the
correlation coefficient is much easier to explain than the covariance.
7-79
Correlation coefficient
Interpretation
r = +1
perfect positive correlation
0 < r < +1
positive linear correlation
r=0
no linear correlation
−1 < r < 0
negative linear correlation
r = −1
perfect negative correlation
Interpretations of Correlation Coefficients
8-79
Significance Test of the Correlation
 Test whether the correlation between the population of two
variables is equal to zero.

H0: ρ=0

t-test
t =
r n-2
1-r
2
, df = n-2

Two-tailed test

Decision rule: reject H0 if +t critical <t, or t<- t critical
9-79
Significance Test of the Correlation
 Example: An analyst is interested in predicting annual sales for XYZ Company,
a maker of paper products. The following table reports a regression of the annual
sales for XYZ against paper product industry sales. The correlation between
company and industry sales is 0.9757. The regression was based on five
observations.
Coefficient
Standard error of the coefficient
Intercept
-94.88
32.97
Slope (industry sales)
0.2796
0.0363
t=
r n2
1 r2

0.9757 3
 7.72
1  0.952
From the t-table, we find that with df = 3 and 95% significance, the two-tailed
critical t-values are ±3.182 (recall that for the t-test the degrees of freedom = n 2). Because the computed t is greater than +3.182, the correlation coefficient is
significantly different from zero.
10-79
Limitations to Correlation Analysis
 Outliers (异常值)

Outliers represent a few extreme values for sample observations.
Relative to the rest of the sample data, the value of an outlier may be
extraordinarily large or small.

Outlier can result in apparent statistical evidence that a significant
relationship exists when, in fact, there is none, or that there is no
relationship when, in fact, there is a relationship.
11-79
Limitations to Correlation Analysis
 Spurious correlation (假相关)

Spurious correlation refers to the appearance of a causal linear
relationship when, in fact, there is no relation. Certain data items may
be highly correlated purely by chance.

That is to say, there is no economic explanation for the relationship,
which would be considered a spurious correlation.
12-79
Limitations to Correlation Analysis
 Nonlinear relationships

Correlation only measures the linear relationship between two
variables, so it dose not capture strong nonlinear relationships
between variables.

For example, two variables could have a nonlinear relationship such
as Y= (1-X) 3 and the correlation coefficient would be close to zero,
which is a limitation of correlation analysis.
13-79
The Basics of Simple Linear Regression
 Linear regression allows you to use one variable to make predictions
about another, test hypotheses about the relation between two variables,
and quantify the strength of the relationship between the two variables.
 Linear regression assumes a linear relation between the dependent and
the independent variables.
 The dependent variable (y) is the variable whose variation is
explained by the independent variable. The dependent variable is
also refer to as the explained variable, the endogenous variable,or
the predicted variable.
 The independent variable (x) is the variable whose variation is used
to explain the variation of the dependent variable. The independent
variable is also refer to as the explanatory variable, the exogenous
variable, or the predicting variable.
14-79
The Basics of Simple Linear Regression
 The simple linear regression model
Yi  b0  b1 X i   i , i  1,..., n
Where,
Yi = ith observation of the dependent variable, Y
Xi = ith observation of the independent variable, X
b0 = regression intercept term
b1 = regression slope coefficient
εi= the residual for the ith observation (also referred to as the disturbance
term or error term)
15-79
The Basics of Simple Linear Regression
 The assumptions of the linear regression

A linear relationship exists between X and Y

X is not random. (even if X is random, we can still rely on the results
of regression models given the crucial assumption that X is
uncorrelated with the error term).

The expected value of the error term is zero (i.e., E(εi)=0 )

The variance of the error term is constant (i.e., the error terms are
homoskedastic)

The error term is uncorrelated across observations (i.e., E(εiεj)=0 for
all i≠j)

The error term is normally distributed.
16-79
Interpretation of regression coefficients
 Interpretation of regression coefficients
 The estimated intercept coefficient ( b̂0 ) is interpreted as the value
of Y when X is equal to zero.
 The estimated slope coefficient ( b̂1 ) defines the sensitivity of Y to
a change in X .The estimated slope coefficient ( b̂1 ) equals
covariance divided by variance of X.
n
Cov( X , Y )
b1 

Var ( X )
(X
i 1
n
17-79
 X )(Yi  Y )
2
(
X

X
)
 i
i 1
b0  Y  b1 X
i
Interpretation of regression coefficients
 Example
 An estimated slope coefficient of 2 would indicate that the dependent
variable will change two units for every 1 unit change in the
independent variable.
 The intercept term of 2% can be interpreted to mean that the
independent variable is zero, the dependent variable is 2%.
18-79
An example: calculate a regression coefficient
 The individual observations on countries' annual average money supply growth
from 1970-2001 are denoted Xi, and individual observations on countries' annual
average inflation rate from 1970-2001 are denoted Y.
Cross-Product
Squared
Deviations
Squared
Deviations
(Xi - X )( Yi- Y )
(Xi - X )2
(Yi - Y)2
0.0676
0.000169
0.000534
0.000053
0.0915
0.0519
0.000017
0.000004
0.000071
New Zealand
0.1060
0.0815
0.000265
0.000156
0.000449
Switzerland
0.0575
0.0339
0.000950
0.001296
0.000697
United Kingdom
0.1258
0.0758
0.000501
0.001043
0.000240
United States
0.0634
0.0509
0.000283
0.000906
0.000088
Sum
0.5608
0.3616
0.002185
0.003939
0.001598
Average
0.0935
0.0603
Variance
0.000788
0.000320
Standard deviation
0.028071
0.017889
Country
Money Supply
Growth Rate
Xi
Inflation
Rate
Yi
Australia
0.1166
Canada
Covariance
19-79
0.000437
An answer: calculate a regression coefficient
n
bˆ 1 =
  Y -Y  X -X 
i
i
i=1
n
  X -X 
2
0.000437
=
= 0.5545, and
0.000788
i
i=1
bˆ 0  Y  bˆ 1 X  0.0603-0.5545(0.0935)  0.0084
Y  0.0084  0.5545 X  
20-79
Standard Error of Estimate & Coefficient of
Determination (R2)
 Standard Error of Estimate (SEE) measures the degree of variability of
the actual Y-values relative to the estimated Y-values from a regression
equation.
 SEE will be low (relative to total variability) if the relationship is very
strong and high if the relationship is weak.
 The SEE gauges the “fit” of the regression line. The smaller the standard
error, the better the fit.
 The SEE is the standard deviation of the error terms in the regression.
21-79
Standard Error of Estimate & Coefficient
Determination (R2)
 The Coefficient Determination (R2) is defined as the percentage of
the total variation in the dependent variable explained by the
independent variable.
 Example: R2 of 0.63 indicates that the variation of the independent
variable explains 63% of the variation in the dependent variable.
22-79
ANOVA Table
 ANOVA Table
df
SS
MSS
Regression
k=1
RSS
MSR=RSS/k
Error
n-2
SSE
MSE=SSE/(n-2)
Total
n-1
SST
-
 Standard error of estimate
SEE 
SSE
 MSE
n2
 Coefficient of determination (R²)


23-79
SSR
SSE
 1
SST
SST
explained variation
unexplained variation

=1total variation
total variation
R2 
For simple linear regression, R²is equal to the squared correlation
coefficient (i.e., R²= r²)
Example:
 An analyst ran a regression and got the following result:
Coefficient
t-statistic
p-value
Intercept
-0.5
-0.91
0.18
Slope
-2.5
10.00
<0.001
df
SS
MSS
Regression
1
7000
?
Error
?
3000
?
Total
41
?
-
 Fill in the blanks of the ANOVA Table.

What is the standard error of estimate?

What is the result of the slope coefficient significance test?

What is the result of the sample correlation?

What is the 95% confidence interval of the slope coefficient?
24-79
Regression coefficient confidence interval
 Regression coefficient confidence interval
bˆ1  t c sbˆ
查表所得
1
 If the confidence interval at the desired level of significance dose not include
zero, the null is rejected, and the coefficient is said to be statistically different
from zero.

sb̂
1
is the standard error of the regression coefficient. As SEE rises,
sb̂
1
also
increases, and the confidence interval widens because SEE measures the
variability of the data about the regression line, and the more variable the data,
the less confidence there is in the regression model to estimate a coefficient.
25-79
Hypothesis Testing about the Regression Coefficient
Significance test for a regression coefficient


H0: b1=The hypothesized value
bˆ1  b1
t
sbˆ
df=n-2
1

Decision rule: reject H0 if +t critical <t, or t<- t critical

Rejection of the null means that the slope coefficient is different from
the hypothesized value of b1
26-79
Predicted Value of the Dependent Variable
 Predicted values are values of the dependent variable based on the
estimated regression coefficients and a prediction about the value of the
independent variable.
 Point estimate
Yˆ  bˆ0  bˆ1 X '
 Confidence interval estimate
Yˆ  t c  s f

t c = the critical t-value with df=n−2
s f = the standard error of the forecast
1 ( X '  X )2
1
( X '  X )2
s f  SEE  1  
 SEE  1  
2
n (n  1) s X
n  ( X i  X )2
27-79
Limitations of Regression Analysis
 Regression relations change over time

This means that the estimation equation based on data from a specific
time period may not be relevant for forecasts or predictions in another
time period. This is referred to as parameter instability.
 The usefulness will be limited if others are also aware of and act on the
relationship.
 Regression assumptions are violated

28-79
For example, the regression assumptions are violated if the data is
heteroskedastic (non-constant variance of the error terms) or exhibits
autocorrelation (error terms are not independent).
Summary of Readings and Framework
SS 3
R9 Correlation and regression
R10 Multiple regression and issues in regression analysis
R11 Time-series analysis
 R12 Excerpt from ’’Probabilistic Approaches: Scenario
Analysis, Decision Trees, and Simulation’’
29-79
Framework
 Multiple Regression
1.
The Basics of Multiple Regression
2.
Interpreting the Multiple Regression Results
3.
Hypothesis Testing about the Regression Coefficient
4.
Regression Coefficient F-test
5.
Coefficient of Determination (R2)
6.
Analysis of Variance (ANOVA)
7.
Dummy variables
8.
Multiple Regression Assumptions
9.
Multiple Regression Assumption Violations
10.
Model Misspecification
11.
Qualitative Dependent Variables
30-79
The Basics of Multiple Regression
 Multiple regression is regression analysis with more than one
independent variable
 The multiple linear regression model
Yi  b0  b1 X 1i  b2 X 2i    bk X ki   i
Xij = ith observation of the jth independent variable
N = number of observation
K = number of independent variables
 Predicted value of the dependent variable
Yˆ  bˆ  bˆ Xˆ  bˆ Xˆ    bˆ Xˆ
0
31-79
1
1
2
2
k
k
Interpreting the Multiple Regression Results
 The intercept term is the value of the dependent variable when the
independent variables are all equal to zero.
 Each slope coefficient is the estimated change in the dependent
variable for a one unit change in that independent variable, holding
the other independent variables constant. That’s why the slope
coefficients in a multiple regression are sometimes called partial
slope coefficient.
32-79
Hypothesis Testing about the Regression Coefficient
 Significance test for a regression coefficient


H0: bj=0
t
bˆ j
sbˆ
df=n-k-1
j
 p-value: the smallest significance level for which the null hypothesis can be
rejected

Reject H0 if p-value<α

Fail to reject H0 if p-value>α
 Regression coefficient confidence interval
ˆ
 b j  tc  sbˆ


33-79
j

Estimated regression coefficient ±(critical t-value) (coefficient standard
error)
Regression Coefficient F-test
 An F-statistic assesses how well the set of independent variables,
as a group, explains the variation in the dependent variable.
 An F-test is used to test whether at least one slope coefficient is
significantly different from zero
H0: b1= b2= b3= … = bk=0
Ha: at least one bj≠0 (j = 1 to k)
 F-statistic
34-79
MSR
F

MSE SSE
SSR
k
(n  k  1)
Regression Coefficient F-test
 Decision rule

reject H0 : if F (test-statistic) > F c (critical value)

Rejection of the null hypothesis at a stated level of significance
indicates that at least one of the coefficients is significantly different
than zero, which is interpreted to mean that at least one of the
independent variables in the regression model makes a significant
contribution on the explanation of the dependent variable.
 The F-test here is always a one-tailed test
 The test assesses the effectiveness of the model as a whole in
explaining the dependent variable
35-79
Coefficient of Determination (R2)
 Interpretation

The percentage of variation in the dependent variable that is collectively
explained by all of the independent variables. For example, an R2 of 0.63
indicates that the model, as a whole, explains 63% of the variation in the
dependent variable.
 Adjusted R2

R2 by itself may not be a reliable measure of the explanatory power of the
multiple regression model. This is because R2 almost always increases as
variables are added to the model, even if the marginal contribution of the
new variables is not statistically significant.
36-79
Analysis of Variance (ANOVA)
 ANOVA Table
d.f.
SS
MSS
Regression
k
RSS
MSR=RSS/k
Error
n-k-1
SSE
MSE=SSE/(n-k-1)
Total
n-1
SST
-
 Standard error of estimate
SSE
 MSE
n  k 1
SEE 
 Coefficient of determination (R²)
RSS
SSE

R2 
 1
SST
SST
 n  1 
2
2 
adjusted R  1  
 1 R 

 n  k  1 

 adjusted R²≤ R²

 adjusted R²may be less than zero
37-79

1  adjusted R 2
1 R2

n 1
n  k 1
Dummy variables
 To use qualitative variables as independent variables in a
regression
 The qualitative variable can only take on two values, 0 and 1
 If we want to distinguish between n categories, we need n−1
dummy variables
38-79
Dummy variables
 Interpreting the coefficients
Example: EPSt = b0 + b1Q1t + b2Q2t + b3Q3t +  t

EPSt = a quarterly observation of earnings per share
Q1t =1 if period t is the first quarter, Q1t =0 otherwise
Q2t =1 if period t is the second quarter, Q2t =0
otherwise
Q3t =1 if period t is the third quarter, Q3t =0
otherwise


39-79
The intercept term, represents the average value of
EPS for the fourth quarter.
The slope coefficient on each dummy variable
estimates the difference in earnings per share (on
average) between the respective quarter (i.e.,
quarter 1, 2, or 3) and the omitted quarter (the
fourth quarter in this case).
y
x1
x2
x3
EPSt
Q1
Q2
Q3
EPS09Q4
0
0
0
EPS09Q3
0
0
1
EPS09Q2
0
1
0
EPS09Q1
1
0
0
EPS08Q4
0
0
0
EPS08Q3
0
0
1
EPS08Q2
0
1
0
EPS08Q1
1
0
0
…
…
…
…
Unbiased and consistent estimator
 An unbiased estimator is one for which the expected value of the
estimator is equal to the parameter you are trying to estimate.

If not, called as unreliable.
 A consistent estimator is one for which the accuracy of the
parameter estimate increases as the sample size increases.
40-79
Multiple Regression Assumptions
 The assumptions of the multiple linear regression






41-79
A linear relationship exists between the dependent and independent
variables
The independent variables are not random ( OR X is not correlated
with error terms). There is no exact linear relation between any two or
more independent variables
The expected value of the error term is zero (i.e., E(εi)=0 )
The variance of the error term is constant (i.e., the error terms are
homoskedastic)
The error term is uncorrelated across observations (i.e., E(εiεj)=0 for
all i≠j)
The error term is normally distributed
Multiple Regression Assumption Violations
 Heteroskedasticity 异方差
 Heteroskedasticity refers to the situation that the variance of the error
term is not constant (i.e., the error terms are not homoskedastic)
 Unconditional heteroskedasticity occurs when the heteroskedasticity
is not related to the level of the independent variables, which means
that it dose not systematically increase or decrease with the change in
the value of the independent variables. It usually causes no major
problems with the regression.
 Conditional heteroskedasticity is heteroskedasticity, that is, variance
of error term is related to the level of the independent variables.
Conditional heteroskedasticity dose create significant problems
for statistical inference.
42-79
Multiple Regression Assumption Violations
 Effect of Heteroskedasticity on Regression Analysis

Not affect the consistency of regression parameter estimators

Consistency: the larger the number of sample, the lower
probability of error.
ˆ
The coefficient estimates (the b j ) are not affected.

The standard errors are usually unreliable estimates.
If the standard errors are too small, but the coefficient
estimates themselves are not affected, the t-statistics will be too
large and the null hypothesis of no statistical significance is
rejected too often (一类错误).
The opposite will be true if the standard errors are too large.
(二类错误)

43-79
The F-test is also unreliable.
Multiple Regression Assumption Violations
 Detecting Heteroskedasticity

Two methods to detect heteroskedasticity
(1) residual scatter plots (residual vs. independent variable)
(2) the Breusch-Pagen χ² test
H0: No heteroskedasticity
BP = n×Rresidual² , df=k one-tailed test 注意：以误差项
squred residuals和X做回归，是此回归的决定系数
 Decision
rule: BP test statistic should be small (χ²分布表）
 Correcting heteroskedasticity

robust standard errors (also called White-corrected standard errors)

generalized least squares
44-79
CFA考试不要求了解如何修正方程，只要
知道如果有异方差问题，用robust
standard error计算t-statistics
Multiple Regression Assumption Violations
 Serial correlation (autocorrelation)
序列相关，自相关

Serial correlation (autocorrelation) refers to the situation that the error
terms are correlated with one another

Serial correlation is often found in time series data

Positive serial correlation exists when a positive regression error in
one time period increases the probability of observing regression
error for the next time period.

Negative serial correlation occurs when a positive error in one period
increases the probability of observing a negative error in the next
period.
45-79
Multiple Regression Assumption Violations
Effect of Serial correlation on Regression Analysis
 Positive serial correlation → Type I error & F-test unreliable

Not affect the consistency of estimated regression coefficients.

Because of the tendency of the data to cluster together from observation to
observation, positive serial correlation typically results in coefficient
standard errors that are too small, which will cause the computed t-statistics
to be larger.

Positive serial correlation is much more common in economic and financial
data, so we focus our attention on its effects.
 Negative serial correlation → Type II error (考试不做要求)

46-79
Because of the tendency of the data to diverge from observation to
observation, negative serial correlation typically causes the standard errors
that are too large, which leads to the computed t-statistics too small.
Multiple Regression Assumption Violations
 Detecting Serial correlation

Two methods to detect serial correlation
 (1) residual scatter plots
 (2) the Durbin-Watson test
H0: No serial correlation
DW ≈ 2×(1−r)

Decision rule
残差的 correlation
思考：当r＝0, 1, -1时候
Reject H0,
Reject H0,
conclude
conclude
positive serial Inconclusive Do not Inconclusive negative serial
reject H0
correlation
correlation
0
d1
dU
4-d1
4
4-dU
47-79
Durbin-Watson test
 H0: No positive serial correlation
DW ≈ 2×(1−r)

48-79
Decision rule
Reject H0,
conclude
positive serial
correlation
Inconclusive Fail to reject null hypothesis of no
positive serial correlation
0
d1
dU
Multiple Regression Assumption Violations
 Methods to Correct Serial correlation

adjusting the coefficient standard errors (e.g., Hansen method): the
Hansen method also corrects for conditional heteroskedaticity.
 The White-corrected standard errors are preferred if only
heteroskedasticity is a problem.

49-79
Improve the specification of the model: The best way to do this is to
explicitly incorporate the time-series nature of the data (e.g., include a
seasonal term).
Multiple Regression Assumption Violations
 Multicollinearity
多重共线性

Multicollinearity refers to the situation that two or more independent
variables are highly correlated with each other

In practice, multicollinearity is often a matter of degree rather than of
absence or presence.

Two methods to detect multicollinearity
(1) t-tests indicate that none of the individual coefficients is
significantly different than zero, while the F-test indicates overall
significance and the R²is high
(2) the absolute value of the sample correlation between any two
independent variables is greater than 0.7 (i.e., ︱r︱>0.7)

50-79
Methods to correct multicollinearity: omit one or more of the
correlated independent variables
Multiple Regression Assumption Violations
 Summary of assumption violations
Assumption violation
Impact
Detection
Conditional
Heteroskedasticity
Type I /II error ① Residual scatter plots
② Breusch-Pagen χ²-test (BP = n×R²)
Positive serial
correlation
Type I error ① Residual scatter plots
② Durbin-Watson test (DW≈2×(1−r))
Multicollinearity
51-79
Type II error ① t-tests: fail to reject H0;
(high R2 and
F-test: reject H0; R²is high
low t-statistics) ② High correlation among
independent variables
Model Misspecification
 There are three broad categories of model misspecification, or ways in which the
regression model can be specified incorrectly, each with several subcategories:

1. The functional form can be misspecified.
 Important variables are omitted.
 Variables should be transformed.
 Data is improperly pooled.

2. Explanatory variables are correlated with the error term in time series
models.
 A lagged dependent variable is used as an independent variable.
 A function of the dependent variable is used as an independent variable
("forecasting the past").
 Independent variables are measured with error.

3. Other time-series misspecifications that result in nonstationarity.
 Effects of the model misspecification: regression coefficients are biased and/or
inconsistent
52-79
Qualitative Dependent Variables
 Qualitative dependent variable is a dummy variable that takes on a value of
either zero or one
 Probit and logit model: Application of these models results in estimates of
the probability that the event occurs (e.g., probability of default).
 A probit model based on the normal distribution, while a logit model is
based on the logistic distribution
 Both models must be estimated using maximum likelihood methods（极
大似然估计）
 These coefficients relate the independent variables to the likelihood of
an event occurring, such as a merger, bankruptcy, or default.
 Discriminant models yields a linear function, similar to a regression equation,
which can then be used to create an overall score, or ranking, for an
observation. Based on the score, an observation can be classified into the
bankrupt or not bankrupt category.
53-79
Summary of Readings and Framework
SS 3
R9 Correlation and regression
R10 Multiple regression and issues in regression analysis
R11 Time-series analysis
 R12 Excerpt from ’’Probabilistic Approaches: Scenario
Analysis, Decision Trees, and Simulation’’
54-79
Framework
 Time-Series Analysis
1.
Trend Models
2.
Autoregressive Models (AR)
3.
Random Walks
4.
Autoregressive Conditional Heteroskedasticity (ARCH)
5.
Regression with More Than One Time Series
6.
Steps in Time-Series Forecasting
55-79
Trend Models
 Linear trend model

yt=b0+b1t+εt

Same as linear regression, except for that the independent variable is
time t (t=1, 2, 3, ……)
yt
t
56-79
Trend Models
 Log-linear trend model

yt=e(b0+b1t)

Ln(yt ) =b0+b1t+εt

Model the natural log of the series using a linear trend

Use the Durbin Watson statistic to detect autocorrelation
57-79
Trend Models
 Factors that Determine Which Model is Best
 A linear trend model may be appropriate if the data points appear to
be equally distributed above and below the regression line (inflation
rate data).
 A log-linear model may be more appropriate if the data plots with a
non-linear (curved) shape, then the residuals from a linear trend
model will be persistently positive or negative for a period of time
(stock indices and stock prices).
 Limitations of Trend Model
 Usually the time series data exhibit serial correlation, which means
that the model is not appropriate for the time series, causing
inconsistent b0 and b1
 The mean and variance of the time series changes over time.
58-79
Autoregressive Models (AR)
 An autoregressive model uses past values of dependent variables as
independent variables
 AR(p) model
xt  b0  b1 xt-1  b2 xt-2  ...  bp xt  p   t


59-79
AR (p): AR model of order p (p indicates the number of lagged
values that the autoregressive model will include).
For example, a model with two lags is referred to as a second-order
autoregressive model or an AR (2) model.
Autoregressive Models (AR)
 Forecasting With an Autoregressive Model
 Chain rule of forecasting

A one-period-ahead forecast for an AR (1) model is determined in the
following manner:



xt 1  b0  b1 xt

Likewise, a two-step-ahead forecast for an AR (1) model is calculated
as:



xt  2  b0  b1 xt 1
60-79
Autoregressive Models (AR)
 Forecasting With an Autoregressive Model, we should prove:

No autocorrelation

Covariance-stationary series

No Conditional Heteroskedasticity
61-79
Autoregressive Models (AR)
 Detecting autocorrelation in an AR model
 Compute the autocorrelations of the residual
 t-tests to see whether the residual autocorrelations differ significantly
from 0,
t  statistics 
 ,
t
t k
1/ n
n is the number of observations in the time series.


62-79
If the residual autocorrelations differ significantly from 0, the model
is not correctly specified, so we may need to modify it (e.g.
seasonality)
Correction: add lagged values
Autoregressive Models (AR)
 Covariance-stationary series

Statistical inference based on OLS estimates for a lagged time series
model assumes that the time series is covariance stationary

Three conditions for covariance stationary
Constant and finite expected value of the time series
Constant and finite variance of the time series
Constant and finite covariance with leading or lagged values

Stationary in the past does not guarantee stationary in the future

All covariance-stationary time series have a finite mean-reverting
level.
63-79
Autoregressive Models (AR)
 Mean reversion



A time series exhibits mean reversion if it has a tendency to move
towards its mean
b0
For an AR(1) model, the mean reverting level is: xt =
(1  b1 )
If
xt 
b0
(1  b1 )
than x t, and if
higher than x t
64-79
the model predicts that x t+1 will be lower
xt 
b0
(1  b1 )
the model predicts that x t+1 will be
Autoregressive model 如果没有
mean reverting level说明follow
random walk.
Autoregressive Models (AR)
 Instability of regression coefficients

Financial and economic relationships are dynamic

Models estimated with shorter time series are usually more stable
than those with longer time series
 So we need to check Covariance stationary
65-79
Compare forecasting power with RMSE
 Comparing forecasting model performance

In-sample forecasts are within the range of data (i.e., time period)
used to estimate the model, which for a time series is known as the
sample or test period.

Root mean squared error (RMSE): the model with the smallest RMSE
is most accurate for out-of-sample
 Out-of-sample forecasts are made outside. In other words, we compare
how accurate a model is in forecasting the y variable value for a time
period outside the period used to develop the model.
66-79
Random Walks
 Random walk

A special AR(1) model with b0=0 and b1=1

Simple random walk: xt =xt-1+εt

The best forecast of xt is xt-1
 Random walk with a drift

xt=b0+b1 xt-1+εt
b0≠0, b1=1

67-79
The time series is expected to increase/decrease by a constant amount
Random Walks
 Covariance stationary

A random walk has an undefined mean reverting level

A time series must have a finite mean reverting level to be covariance
stationary

A random walk, with or without a drift, is not covariance stationary

The time series is said to have a unit root if the lag coefficient is equal
to one
Dickey and Fuller test
68-79
Random Walks
 The unit root test of nonstationarity
 The time series is said to have a unit root if the lag coefficient is equal to
one
 A common t-test of the hypothesis that b1=1 is invalid to test the unit
root
 Dickey-Fuller test (DF test) to test the unit root
Start with an AR(1) model xt=b0+b1 xt-1+εt
Subtract xt-1 from both sides xt-xt-1 =b0+(b1 –1) xt-1+εt
xt-xt-1 =b0+g xt-1+εt
H0: g=0 (has a unit root and is nonstationary) Ha: g<0 (does not
have a unit root and is stationary)
Calculate conventional t-statistic and use revised t-table
If we can reject the null, the time series does not have a unit root and
is stationary
69-79
Random Walks – if a time series appears to have a unit
root
 If a time series appears to have a unit root, how should we model
it ???
 One method that is often successful is to first-difference the time
series (as discussed previously) and try to model the firstdifferenced series as an autoregressive time series
 First differencing
e.g. 2, 5, 10, 17, ? , 37

Define yt as yt = xt - xt-1

This is an AR(1) model yt = b0 + b1 yt-1 +εt , where b0=b1=0

The first-differenced variable yt is covariance stationary
70-79
Autoregressive Models (AR)
 Seasonality –a special question

Time series shows regular patterns of movement within the year

The seasonal autocorrelation of the residual will differ significantly
from 0

We should uses a seasonal lag in an AR model

For example: xt=b0+b1 xt-1+ b2 xt-4+εt
71-79
Autoregressive Conditional Heteroskedasticity (ARCH)
 Heteroskedasticity refers to the situation that the variance of the
error term is not constant
 Test whether a time series is ARCH(1)


多元回归中用BP test
 t2  a0  a1 t21  ut
If the coefficient a1 is significantly different from 0, the time series is
ARCH(1)
 Generalized least squares must be used to develop a predictive
model
 Use the ARCH model to predict the variance of the residuals in
following periods
72-79
Regression with More Than One Time Series
 In linear regression, if any time series contains a unit root, OLS may be
invalid
 Use DF tests for each of the time series to detect unit root, we will have 3
possible scenarios

None of the time series has a unit root: we can use multiple regression

At least one time series has a unit root while at least one time series does
not: we cannot use multiple regression

Each time series has a unit root: we need to establish whether the time
series are cointegrated.
If conintegrated, can estimate the long-term relation between the two
series (but may not be the best model of the short-term relationship
between the two series).
73-79
Regression with More Than One Time Series
 Use the Dickey-Fuller Engle-Granger test (DF-EG test) to test the
cointegration

H0: no cointegration

If we cannot reject the null, we cannot use multiple regression

If we can reject the null, we can use multiple regression
74-79
Ha: cointegration
Summary of Readings and Framework
SS 3
R9 Correlation and regression
R10 Multiple regression and issues in regression analysis
R11 Time-series analysis
 R12 Excerpt from“Probabilistic Approaches: Scenario
Analysis, Decision Trees, and Simulation”
75-79
Simulation
 Steps in Simulation

Determine “probabilistic” variables

Define probability distributions for these variables
 Historical data
 Cross sectional data
 Statistical distribution and parameters

Check for correlation across variables
 When two variables are strong correlated, one solution is to pick only
one of the two inputs; the other is to build the correlation explicitly into
the simulation.

76-79
Run the simulation
Simulation
 Advantage of using simulation in decision making

Better input estimation

A distribution for expected value rather than a point estimate
 Simulations with Constraints

Book value constraints
 Regulatory capital restrictions

Financial service firms
 Negative book value for equity

Earnings and cash flow constraints
 Either internally or externally imposed

Market value constraints
 Model the effect of distress on expected cash flows and discount rates
77-79
Simulation
 Issues in using simulation

GIGO

Real data may not fit distributions

Non-stationary distributions

Changing correlation across inputs
78-79
Comparing the Approaches
 Choose scenario analysis, decision trees, or simulations

Selective versus full risk analysis

Type of risk
Discrete risk vs. Continuous risk
Concurrent risk vs. Sequential risk

Correlation across risk
Correlated risks are difficult to model in decision trees
Risk type and Probabilistic Approaches
79-79
Discrete/
Continuous
Correlated/
Independent
Sequential/
Concurrent
Risk approach
Discrete
Correlated
Sequential
decision trees
Discrete
Independent
Concurrent
scenario
analysis
Continuous
Either
Either
simulations