Regression Analysis Overview for Finance

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What does the symbol α̂ represent in the context of the simple regression example?

The symbol α̂ represents the intercept of the regression equation.

How is the expected return on fund XXX calculated using the regression equation?

The expected return is calculated by substituting the expected market return into the regression equation.

What is the significance of the coefficient β̂ in the regression equation?

The coefficient β̂ indicates the sensitivity of the fund's return to changes in the market return.

If an analyst expects the market to yield a 20% return next year, what fundamental expectation does this imply for risk and reward?

It implies that investors anticipate higher rewards associated with risks taken in equities. Signup and view all the answers

Can the fitted regression line ŷt = -1.74 + 1.64xt predict negative returns for fund XXX? Explain.

Yes, if the market excess return is sufficiently negative, the model can predict negative returns. Signup and view all the answers

What are the key steps to follow before performing regression analysis?

Identifying research questions, determining data needs, finding appropriate databases, and analyzing variable characteristics are key steps. Signup and view all the answers

What is the role of the OLS estimator in regression analysis?

The OLS estimator provides the best linear unbiased estimates of the regression coefficients. Signup and view all the answers

What assumptions underlie the Classical Linear Regression Model (CLRM)?

Key assumptions include linearity, independence of errors, homoscedasticity, and normality of error terms. Signup and view all the answers

How can one distinguish between regression and correlation?

Regression establishes a functional relationship between variables, while correlation measures the strength and direction of their linear relationship. Signup and view all the answers

What is the significance of precision and standard errors in regression analysis?

Precision indicates the accuracy of the coefficient estimates, while standard errors measure the variability of these estimates. Signup and view all the answers

What is a regression model with dummy variables used for?

Dummy variables are used to include categorical data in regression models, allowing for the analysis of qualitative variables. Signup and view all the answers

What are the properties of the OLS estimator?

The OLS estimator is linear, unbiased, efficient, and satisfies the Gauss-Markov theorem's conditions. Signup and view all the answers

In the context of simple regression, what is meant by the term 'nonlinear regression models'?

Nonlinear regression models describe relationships that cannot be accurately represented as linear. Signup and view all the answers

What is the significance of the interval [0.99, 1.01] in the context of OLS estimates?

The interval [0.99, 1.01] indicates that the OLS estimates are statistically consistent with probability values close to 1. Signup and view all the answers

How does Figure 5.2 contribute to understanding the precision of OLS estimates?

Figure 5.2 provides graphical representation, illustrating specific ranges that indicate the precision of the OLS estimates. Signup and view all the answers

What role do the values 1.52 and 1.73 play in the interpretation of Figure 5.4?

The values 1.52 and 1.73 represent the critical points for understanding how the estimates span across the confidence intervals. Signup and view all the answers

What implications arise from the negative intervals seen in Figure 5.4, such as [-1.57, 3.39]?

Negative intervals like [-1.57, 3.39] imply potential instability or uncertainty in the estimates within that range. Signup and view all the answers

In the context of precision, why is an interval like [0.70, 1.38] considered important?

This interval indicates a range of values where the estimates may vary significantly, affecting overall confidence levels. Signup and view all the answers

How do the values 0.91 and 0.89 in relation to precision reflect on the reliability of the estimates?

Values of 0.91 and 0.89 suggest a tight clustering of estimates indicating high reliability and low variance. Signup and view all the answers

What can be inferred from the presence of both positive and negative ranges in the estimated intervals?

The presence of both positive and negative ranges indicates that the estimates encapsulate a wide spectrum of uncertainty. Signup and view all the answers

What does the interval [0.59, 1.49] indicate about the variability of the estimates?

This interval shows significant variability which may challenge the reliability of interpretations made from the estimates. Signup and view all the answers

What does the term ∑ xt2 measure in the context of standard errors?

It measures the dispersion of data points from the y-axis. Signup and view all the answers

In a simple regression model, what are the unknown parameters that regression analysis aims to estimate?

The parameters ε (error term) and β (slope coefficient). Signup and view all the answers

What does the notation ε̂ and β̂ represent in regression analysis?

They represent the estimated values of ε and β, respectively. Signup and view all the answers

In the multiple regression example provided, list two independent variables that influence the sales price of a house.

Lot size and number of bedrooms. Signup and view all the answers

What is the significance of OLS estimation in the context of multiple regression?

OLS estimation is used to minimize the sum of squared differences between observed and predicted values. Signup and view all the answers

What is represented by the dependent variable yi in the multiple regression example?

The sales price of the house. Signup and view all the answers

How does multiple regression differ from simple regression?

Multiple regression involves more than one independent variable while simple regression uses only one. Signup and view all the answers

Why is it important to consider several factors like bathrooms and storeys in a multiple regression model for housing prices?

These factors can significantly impact the value of the housing market and provide a more accurate estimation. Signup and view all the answers

Why does OLS estimation involve taking vertical deviations rather than horizontal distances?

OLS estimation focuses on minimizing the sum of squared vertical distances since this reflects the error between the observed and predicted values. Signup and view all the answers

Why are the vertical distances squared before being added in OLS estimation?

The vertical distances are squared to eliminate any negative values and to give greater weight to larger deviations, highlighting significant errors in predictions. Signup and view all the answers

What are the five assumptions usually made about the unobservable error terms in the classical linear regression model (CLRM)?

The five assumptions are: errors have a mean of zero, errors are uncorrelated, errors have constant variance, errors are normally distributed, and the regressors are not perfectly collinear. Signup and view all the answers

Explain the meaning of 'errors have a mean of zero' in the context of CLRM.

This assumption means that the expected value of the error terms is zero, indicating that on average, the errors do not systematically overstate or understate the true relationship. Signup and view all the answers

What differentiates the OLS estimator from other estimators?

The OLS estimator is superior because it minimizes the sum of squared residuals, which leads to optimal estimates under the assumptions of the linear regression model. Signup and view all the answers

Why is it important for regression models using OLS to be linear in the parameters?

It is important because linearity ensures that the relationship between independent and dependent variables can be easily estimated and interpreted. Signup and view all the answers

How does ensuring that errors are uncorrelated contribute to the effectiveness of the OLS estimator?

Uncorrelated errors imply that the errors do not exhibit systematic patterns relative to one another, which supports the assumption of independence in modeling. Signup and view all the answers

Why is it necessary to check for perfect multicollinearity among regressors in OLS models?

Perfect multicollinearity means that one or more regressors can be perfectly predicted by others, making it impossible to estimate their individual effects accurately. Signup and view all the answers

What does a confidence interval represent and why is it important in statistical inference?

A confidence interval represents the range of values within which the true parameter is likely to fall, providing a measure of uncertainty in an estimate. Signup and view all the answers

In the hypothesis test H0: B = 1 versus H1: B ≠ 1, what does the alternative hypothesis signify?

The alternative hypothesis signifies that the parameter B is not equal to 1, indicating a significant effect or difference exists. Signup and view all the answers

Calculate the standard error (SE) of the estimate from the regression output provided.

The standard error of the estimate is 0.2561. Signup and view all the answers

Discuss the effect of sample size on the confidence interval width and its implications on statistical inference.

As sample size increases, the confidence interval tends to become narrower, leading to more precise estimates and stronger conclusions. Signup and view all the answers

How would you interpret the confidence interval [-1.33, 4.36] in Figure 5.4?

This confidence interval suggests that we are reasonably confident the true parameter lies between -1.33 and 4.36. Signup and view all the answers

What does a p-value indicate in the context of hypothesis testing?

A p-value indicates the probability of observing the data, or something more extreme, assuming the null hypothesis is true. Signup and view all the answers

Identify two factors that can influence the accuracy of OLS (Ordinary Least Squares) estimates.

Multicollinearity and heteroscedasticity are two factors that can influence the accuracy of OLS estimates. Signup and view all the answers

Why is it important to consider both the test of significance and confidence intervals when testing hypotheses?

Considering both allows for a more comprehensive understanding of the results, as significance tests indicate if an effect exists while confidence intervals show the magnitude and uncertainty. Signup and view all the answers

Study Notes

Classical Linear Regression Model (CLRM): Overview

Classical Linear Regression Model (CLRM) is a statistical method used to understand the relationship between variables.
It assumes that there's a linear relationship between dependent and independent variables. Also assumes that the errors are normally distributed.
It is a core method in financial research.

Data Handling Review

Research questions need to be clearly defined.
The type of data required to answer the research questions should be identified.
Suitable data sources (databases) for collecting the necessary data.

Agenda Topic 2

Definition of regression model.
Simple regression and theory including the OLS estimates.
Nonlinear regression models.
Classical linear regression model (CLRM) assumptions.
OLS estimator's properties: Precision and standard errors.
Multiple regression including dummy variables.

Course Material Topic 2

Textbook reading: Brooks (2019), Chapter 3, subsections 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, Chapter 4, subsections 4.1, 4.6.
Background reading: Koop (Analysis of Financial Data), Chapters 4, 5, 6, 7; Koop (Introduction to Econometrics), Chapters 3, 4, 5.

Regression Model

Regression model is a method used to understand the relationship between variables.
General: Aims to describe and evaluate the relationship between a given variable and one or more other variables.
Specific: Attempts to explain the movements in one variable by reference to movements in one or more other variables.

Regression Model (Continued)

Denotes the dependent variable by 'y' and independent variable(s) by 'x1, x2...xk' where there are 'k' independent variables.
Simple regression: A situation where y depends on only one x variable (ie. k=1).
Regression vs. Correlation: Regression analysis treats the dependent and independent variables differently. In the correlation analysis, the variables are treated symmetrically.

Simple Regression: Introduction

Used to understand the relationships between two variables.
A simple regression is a best fitting line through XY plot points, capturing the relationship. The line of best fit should minimise the sum of squared residuals.

Simple Regression: Theory

Finding a straight line that best fits the data by using the general equation for a straight line (y = a + βx).
'a' represents the intercept of the line.
'β' represents the slope of the line.
The equation is deterministic without any random disturbance.
An error term u is added to the equation to account for real world data (Yt = a + βxt + Ut)

Simple Regression: Theory (Continued)

The disturbance/error term is present to represent missing variables or measurement error.

Simple Regression Model

The dependent variable (y) is to be estimated based on the independent variable(x);
What we know: yt and xt
What we don't know: α, β or ut.

Distinction Between Disturbance Terms and Residuals

True Regression Line (Population Regression Function): Yt = a + βxt + Ut
ut = disturbance/error term
Estimated Regression Line (Sample Regression Function): Yt = â + βˆxt + ût
ût = residual/ error

The Population versus the Sample

Population = the complete group of objects or people under study.
Sample = a selected portion of the population.
Random sample = every member of the population has an equal chance of being selected.

How do we choose â and ß?

We want to find a best fit line that will minimize errors in the model
The one that minimizes the sum of squared residuals or use the ordinary least square (OLS) to find.

Derivation of OLS Estimator

Using given data to estimate the values of a and ß.
Solution: β and â calculations formulas.

Jargon of Regression

Explanation of terms for regression model, including dependent variable, explanatory variable, coefficients and estimates.

Interpretation of OLS Estimates

Estimating the value of the dependent variable (y) if the independent variable is equal to zero.
More relevant to understand the marginal effect on y for a unit change in x.
The accuracy of the intercept estimate is subject to the available observations (or data) close to the y-axis.

Simple Regression Example

Data on excess returns and excess market returns for a fund manager.
Illustration of use case where the beta (in CAPM) framework on this fund is positive relating the returns from the fund to returns in the market index, hence forming a scatterplot.

Simple Regression Example (Continued)

OLS formulae yields estimates a and β.

Time for a break!

Example: Data (Topic 2)

Stock Data. (Monthly, End-of-Month).
Data covering a period of January 2002 to February 2018.
Source: Refinitiv Datastream
Includes stock prices of S&P500, Ford, General Electric (GE), Microsoft, Oracle. Includes a US Risk-free rate.

Simple Regression: Model

Goal: Explain movements in the excess return of a give asset, say Ford's stock, by reference to movements in the S&P500 excess return.
The regression equation takes the form of: (Ford-risk-free) = a + β (S&P500 market return- risk-free) + error

Simple Regression: Output (Topic 2)

Output table of the regression result.

Multiple Choice Question (Wooclap)

Multiple choice questions related to topics for the student to test knowledge..

Nonlinear Regression Models

So far regressions of yt on xt
Linear in the parameters (a and β). Linear in the parameter means that the parameters are not multiplied by each other. (ie. not added, divided, squared etc.) It does not have to be linear for the variables.

Nonlinear Regression Models (Continued)

Regression equation in double logarithmic form: In Yt = a + β In Xt + Ut
Coefficients can be interpreted as elasticities • dyt/dx = dln Yt/ dlnXt, = Y0/X0

Non-linear Regression Models (Continued)

How might you know if relationship is nonlinear? • Consulting financial theory or using theoretical insights carefully examining of XY-plots or residual plots or hypothesis testing

Classical Linear Regression Model Assumptions

Model widely used in classical, normal linear regression or CLRM)
We observe data for xt, but since yt also depends on ut, we must be specific about how the ut are generated. We usually make the following assumptions about Ut: • E(ut) = 0 • Var(ut) = σ2 • Cov (ui, uj) = 0 ( i ≠ j) • Cov(Ut, Ut) = 0 ( for t≠s) • Ut is normally distributed

Classical Linear Regression Model Properties

OLS estimator is unbiased.
OLS estimator is efficient (relative to other estimators).
OLS estimator has a normal distribution.
Used for hypothesis testing.

Consistency versus Unbiasedness

Consistent: OLS estimators → ∞ → a and ẞ are consistent, ie. estimates will converge to the true values of a and ẞ, as the sample size increases
Unbiased: OLS estimates of a and ẞ are unbiased. E(â) = a and E(ẞ) = β

Unbiasedness versus Efficiency

OLS estimator has the smallest variance (ie, efficient) among the class of linear unbiased estimators.

Precision and Standard Errors

â and β are only estimates (sample specific).

Precision of OLS Estimates

Needs a measure of accuracy, reliability, or precision of the estimations (â and β).
The precision of the estimative is given by its standard error. SE(α) = S √ [∑X²t / ∑(Xt − X)² ] SE(β) = S √[1 / ∑(Xt − X)² ]

Factors Affecting Precision of OLS Estimates

More precise estimates if there are more data points.
Less scattering/variability (ie. less variability in residuals, 's' is lower)
More variability in 'X' (data points spread out on the x-axis)

Jargon of Regression (Important Terms - continued)

Dependent variable and explaonatory variables
Coefficients and estimates
Example for better understanding (case of house prices)

Interpretation of OLS Estimates (Important Terms - Continued)

Accuracy of the intercept.

Simple Regression: Example

Simple regression example based on excess returns (daily data) of a particular financial security/investment from a fund manager.

Time for a break!

Example: Data (Topic 3)

Stock data (monthly, end-of-month).
Dates starting from January 2002 up to February 2018.
Source: Refinitiv Datastream.
Includes stock prices, and the US risk-free rate.

R Example (Topic 3)

The data is for regression.
The regression model is to test if the returns of the capital assets are equal to one.

Hypothesis Testing: Test of Significance Approach

Assume the regression equation takes a particular form.
Steps of conducting a test of significance:
- Estimate the coefficients â and β and their standard errors (SE(â), SE(β)).
- Calculate the test statistic.
- Obtain the critical value from table t (at a given alpha level).
- Perform the test.

Hypothesis Testing: Confidence Interval Approach

Understand uncertainty about the precision of estimates.
Interval estimate versus point estimate (calculate the interval estimate).
Examples of confidence intervals for data sets:
- Figure 5.1: Very small sample size
- Figure 5.2: Large sample size, large error variance
- Figure 5.3: Large sample size, small error variance
- Figure 5.4: Limited range of x values

Hypothesis Testing: Example

Use regression results to test the null hypothesis the beta is equal to 1 against a two-sided alternative.
Obtain critical values at a given significance level.
Perform the test using t statistics.

Testing Other Hypotheses

Testing the hypothesis that B=0 or B=2.
Use confidence intervals to test these cases.
Decision rules for testing hypotheses.

Changing the Size of the Test

A test of significance approach will yield different conclusions for tests if different significance levels are used (eg, 5% vs 10%). This is especially notable in marginal cases where the outcome depends heavily on the size of the test.

A Special Type of Hypothesis Test: The t-ratio

Recall the formula for a test of significance using a t-test.
Hypothesis is of a population coefficient being zero against a two-sided alternative. This is called the t-ratio test
T-ratio (t-statistic): The ratio of a coefficient to its standard error.

The t-ratio: An Example

Illustrative data with coefficient estimates, standard errors and t-ratios for an intercept and slope coefficients.
Check to see if the coefficients are significantly different from zero.
The null hypothesis is to be rejected if the t-ratio lies outside the acceptance region.

What does the t-ratio tell us?

If the null hypothesis is rejected, the result is significant; in case of non-significance, the explanatory variable does not help to justify the model
In practice, it is often a good statistical reason to always keep a constant term even when it's not statistically significant.

Some More Terminology

Explanation of the meaning of statistically significant result and how it relates to practical significance.

The Exact Significance Level (p-value)

Explanation of the p-value in terms of choosing critical values and the possibility of rejecting the null hypothesis.

R Example (Topic 3)

The regression model must be linear in the parameters for OLS to be a valid estimation technique.

Multiple Regression: Definition

This study is concerned with situations where a dependent variable (y) depends on more than one independent variable.

Multiple Regression (continued): Estimation

Multiple regression model.
Calculation of OLS estimates (β1, β2, β3, ...βk).
Minimization of the residual sum of squares (RSS)

Multiple Regression: OLS Estimates

Mathematical reasoning and formulation of the OLS estimate calculation formula.

Multiple Regression: Interpretation of OLS Estimates

Mathematical/verbal intuition to interpret the coefficients of OLS estimations.
How to interpret the partial effect (ceteris paribus).

Example: Explaining House Prices

Explain the fitted regression line (regression results).
How to interpret specific coefficients in the multiple regression equation.

Multiple Regression R Simple Regression Model

Specify the goal of the regression model.
Formulate the regression equation.

Multiple Regression R: Output (Topic 2)

• Detailed output table of regression results.

Multiple Choice Question (Wooclap)(Topic 2 and 3)

Multiple choice questions covering the topics and testing the understanding of the topics.

Regression with Qualitative/Dummy Variables

Definition of Dummy Variables
Intercept dummies
Slope dummy variables.
Explanation of the example and how it relates to the concepts or theory.
Example relating to the salary of a Trader in which the annual salary depends on age, sex, location, years of work and education qualifications.

Self-Study

Brief overview of the three types of self-study activities (Multiple Choice Questions(Wooclap), Self-study Questions(Textbook), PC Exercises with R). Explain the importance of these activities in enhancing the learning process.

Multiple Choice Question (Wooclap)(Topic 3 and 4)

Multiple choice questions related to the topics testing knowledge of the topics.

Classical Linear Regression Model: Assumptions + Properties

Assumptions about disturbances, including :

E(ut) = 0
Var(ut) = σ2
Cov(ut, uj) = 0 when t ≠ s
Cov(ut, Xj) = 0
Ut~ normally distributed

Classical Linear Regression Model Violations or Pitfalls :

1.E(Ut) ≠0 (Missing constant term)
1. Heteroskedasticity
3.Residual autocorrelation
4.Omitted variable bias
1. Ut is not normally distributed.

Heteroskedasticity Detection

Graphical methods: Scatter plots, residual graphs
Formal tests: White´s test.
Graphical methods: Time series plot of residuals versus time

Heteroscedasticity Detection

Graphical methods: Scatter plot of residuals versus the independent variables to check whether the error variances change with levels of the independent variables
Formal tests: White's test (using an auxiliary regression), or other tests such as Goldfeld-Quandt or Breusch-Pagan tests • Consider a regression: (R

R ) = a + β(R

R ) + ut t Ford t f SP500 t f • Goal: Test whether there is heteroscedasticity (with formal tests).

Heteroscedasticity: GLS

If the form of heteroscedasticity is known, then we can use GLS (generalised least squares)
A simple illustration for GLS; suppose, the error's variance is related to another variable zt, var(ut) = σ² zt
To remove heteroskedasticity from the regression, divide the equation by zt.
Estimate the model with GLS.

Heteroscedasticity: Solution 2-White SE

Use White (robust) standard errors as a solution, which is easy to implement in R.

Residual Autocorrelation: Detection

Graphical methods: Time series plot of residuals, scatter plot
Formal tests: Durbin-Watson (DW) test for first-order autocorrelation; Breusch-Godfrey test for higher-order autocorrelation; Durbin's 'h' test

Residual Autocorrelation: Detection (Continued)

• Computing Durbin-Watson h, using t-tables.

Residual autocorrelation: Detection (Continued)

• Conclusion: No evidence of residual autocorrelation at 5% significance level.

Durbin-Watson Test

Formal statistical test.
Residual autocorrelation of first-order: Cov(Ut, Ut−1) ≠ 0.

Durbin-Watson Test: Critical Values

Table of critical values for DW test at 5 % significance for different values of n and k.

Dynamic Models

Introduction to static vs dynamic models.

Dynamic Models: Example

Data on monthly data for 5 years and the price of oil (dollars per barrel)
Regression model (y in ($000) & x in dollars per barrel).
Use cases where lagged explanatory variables might be relevant.

Dynamic Models: Example (Continued)

• Interpretations of different coefficients in different periods in the dynamic model, ceteris paribus.

Dynamic Models: Problems

• What does a regression with very many lags of the same variable actually mean? • You lose data. How may the amount of information vary when using lagged variables vs no lagged variables. • How do you choose the lag length? (Financial theory and t-tests)

Dynamic Models: First Differences

Switching to a model in first difference can help with the problem of autocorrelation.
Denote the first difference using ∆

Omitted Variables Bias: Introduction

Omission of an important explanatory variable creates a bias.
The estimated coefficients on all the other variables will be biased and inconsistent unless the omitted variable is uncorrelated with all the other included variables.
Even if the excluded variable is not correlated, the constant term will still be biased.

Omitted Variables Bias: Example 1

Use cases relating to house prices where the number of bedrooms was the independent variable but other characteristics like size, number of bathrooms and storeys were omitted.

Omitted Variables Bias: Example 2

Use cases in finance. Relating to Capital Asset Pricing Models
Explain and motivate the model.

Non-Normality: Introduction

Assumptions about disturbances, including the normality assumption
Why we need to assume normality for hypothesis testing.
Testing the normality assumption: The Bera Jarque normality test

Non-Normality: Detection

Use the Bera Jarque normality test in R to assess if the errors (or disturbances) in the regression model can be considered normally distributed.

Non-Normality: Remove Outliers

Remove outlier using dummy variable (time dummy) for that period, re-estimate model (including the dummy) and test for normality again.

Some Exercises (Topic 6)

Illustrative problem to solve. Suggest possible issues (e.g multicollinearity or omitted variable bias) and how to improve the model for a better estimation.

Time Series: Non-Stationarity and Spurious Regression

Spurious regression arises when y and x are non-stationary time series in regression; it should only be run when x and y are stationary time series or are co-integrated.

Time Series: Non-Stationarity Definition

Non-stationary time series: time series which don't have constant mean or variance over time.

Time Series: Non-Stationarity (Continued)

Deterministic trend process
Random walk model with drift

Time Series: Unit Root Process

Different forms of unit root (DF or Augmented DF).
How and why we need to scale a normal to have a zero mean and a unit variance.

Time Series: Non-stationarity: Example: Stock Prices

Illustrative graph showing stochastic and non-stationary time series data, in log form, and how the nature or trend will change after differencing.

Time Series: Differencing

∆yt = yt -yt−1.
Differencing measures the change from one period to the next.
The difference in values is often used to reduce issues such as dependence in an Autoregressive model.

Time Series: Unit Root Testing: Dickey-Fuller Test

The approach to identify whether a time series contains unit roots using regression tests with specific forms of the model:
Dickey-Fuller test (DF test)
Augmented Dickey-Fuller test (ADF test).

Time Series: Unit Root Testing: Dickey-Fuller Test (Continued)

What and how the t-statistic should be defined in the DF test to test for a unit root.
Which tests can be run against the null-hypotheses and the associated critical values..

Event Study in R: Example

Goal: analyze whether a rating downgrade from a particular company generates a significant market reaction in the stock market.

Event Study in R: Example (Continued)

• Event definition: • Event definition • Event and estimation window • Regression model/normal performance • Abnormal return calculation • Aggregation of abnormal returns • Hypothesis testing for significance.

Panel Data

Financial datasets which contains cross-sectional and time series dimensions, eg.
Survey of a large number of firms or countries over time.
How panel data can solve the problems with simple regression models, and panel data are more complex to estimate, but they are also more robust.

Panel Data: Problems with Pooled Model

Using pooled model is not appropriate since different individuals (firms, counties) have different regression lines and these are ignored.

Problem with Fixed Effects Model with Dummy Variables

When number of individuals is very large (n=1000) in panel regression. Regression with huge number of dummies
Losing observations for statistical analysis (low degrees of freedom) -Solution:De-mean the variables

Random Effects Model

Random effects model does not use dummy variables but assumes the individual effect is a random variable, ie: Yit = a¡ + βxit + wit, where Wit = €¡ + Vit
Advantages relatively to fixed effects model:
Only one intercept (no need for multiple dummies).
Less coefficients will need to be estimated, leading to more efficient estimates.
Allows to control time-invariant explanatory variables.

Random Effects Model: Estimation

Estimating the Random effects model by OLS (using robust standard errors), and the problem of choosing between models in these cases using the Hausman test, p<0.05 reject pooled model but p=0.9283 prefer pooled model.

Fixed or Random Effects Model: Hausman Test

How and why use the hausman test (Ho: Cov(xit, wit)=0), using appropriate p-value and deciding which model is suitable for the data.

Fixed or Random Effects Model: Final Remarks

Additional guidelines when choosing among fixed and or random estimation models.
Fixed effects implicitly controls time-invariant variable; random effects model can also estimate time-invariant variables but only the ones that are explicit variables in the model.
Summary (of various topics)*
Summary of various topics in the course providing a concise overview of the core concepts, and how they should apply to different situation of data.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

This quiz covers essential concepts in regression analysis, focusing on its application in finance. Topics include the meaning of coefficients, the roles of estimators, assumptions of the Classical Linear Regression Model, and the distinction between regression and correlation. Perfect for students wanting to deepen their understanding of financial regression techniques.

Regression Analysis Overview for Finance

Choose a study mode

Podcast

Questions and Answers

What does the symbol α̂ represent in the context of the simple regression example?

How is the expected return on fund XXX calculated using the regression equation?

What is the significance of the coefficient β̂ in the regression equation?

If an analyst expects the market to yield a 20% return next year, what fundamental expectation does this imply for risk and reward?

Can the fitted regression line ŷt = -1.74 + 1.64xt predict negative returns for fund XXX? Explain.

What are the key steps to follow before performing regression analysis?

What is the role of the OLS estimator in regression analysis?

What assumptions underlie the Classical Linear Regression Model (CLRM)?

How can one distinguish between regression and correlation?

What is the significance of precision and standard errors in regression analysis?

What is a regression model with dummy variables used for?

What are the properties of the OLS estimator?

In the context of simple regression, what is meant by the term 'nonlinear regression models'?

What is the significance of the interval [0.99, 1.01] in the context of OLS estimates?

How does Figure 5.2 contribute to understanding the precision of OLS estimates?

What role do the values 1.52 and 1.73 play in the interpretation of Figure 5.4?

What implications arise from the negative intervals seen in Figure 5.4, such as [-1.57, 3.39]?

In the context of precision, why is an interval like [0.70, 1.38] considered important?

How do the values 0.91 and 0.89 in relation to precision reflect on the reliability of the estimates?

What can be inferred from the presence of both positive and negative ranges in the estimated intervals?

What does the interval [0.59, 1.49] indicate about the variability of the estimates?

What does the term ∑ xt2 measure in the context of standard errors?

In a simple regression model, what are the unknown parameters that regression analysis aims to estimate?

What does the notation ε̂ and β̂ represent in regression analysis?

In the multiple regression example provided, list two independent variables that influence the sales price of a house.

What is the significance of OLS estimation in the context of multiple regression?

What is represented by the dependent variable yi in the multiple regression example?

How does multiple regression differ from simple regression?

Why is it important to consider several factors like bathrooms and storeys in a multiple regression model for housing prices?

Why does OLS estimation involve taking vertical deviations rather than horizontal distances?

Why are the vertical distances squared before being added in OLS estimation?

What are the five assumptions usually made about the unobservable error terms in the classical linear regression model (CLRM)?

Explain the meaning of 'errors have a mean of zero' in the context of CLRM.

What differentiates the OLS estimator from other estimators?

Why is it important for regression models using OLS to be linear in the parameters?

How does ensuring that errors are uncorrelated contribute to the effectiveness of the OLS estimator?

Why is it necessary to check for perfect multicollinearity among regressors in OLS models?

What does a confidence interval represent and why is it important in statistical inference?

In the hypothesis test H0: B = 1 versus H1: B ≠ 1, what does the alternative hypothesis signify?

Calculate the standard error (SE) of the estimate from the regression output provided.

Discuss the effect of sample size on the confidence interval width and its implications on statistical inference.

How would you interpret the confidence interval [-1.33, 4.36] in Figure 5.4?

What does a p-value indicate in the context of hypothesis testing?

Identify two factors that can influence the accuracy of OLS (Ordinary Least Squares) estimates.

Why is it important to consider both the test of significance and confidence intervals when testing hypotheses?

Study Notes

Classical Linear Regression Model (CLRM): Overview

Data Handling Review

Agenda Topic 2

Course Material Topic 2

Regression Model

Regression Model (Continued)

Simple Regression: Introduction

Simple Regression: Theory

Simple Regression: Theory (Continued)

Simple Regression Model

Distinction Between Disturbance Terms and Residuals

The Population versus the Sample

How do we choose â and ß?

Derivation of OLS Estimator

Jargon of Regression

Interpretation of OLS Estimates

Simple Regression Example

Simple Regression Example (Continued)

Time for a break!

Example: Data (Topic 2)

Simple Regression: Model

Simple Regression: Output (Topic 2)

Multiple Choice Question (Wooclap)

Nonlinear Regression Models

Nonlinear Regression Models (Continued)

Non-linear Regression Models (Continued)

Classical Linear Regression Model Assumptions

Classical Linear Regression Model Properties

Consistency versus Unbiasedness

Unbiasedness versus Efficiency