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Market Structure, Trading, and LiquidityFIN 2340

Dr. Michael Pagano, CFA

Econometric Topics

Adapted and Excerpted from Slides by:

Dr. Ian W. Marsh

Cass College, Cambridge U. and CEPR

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(1) Two-Variable Regression: Estimation & Hypothesis Testing

(2) Extensions of the Two-Variable Model: Functional Form

(3) Estimating Multivariate Regressions

(4) Multivariate Regression Inference Tests & Dummy Variables

Overview of Key EconometricTopics

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Introduction

• Introduction to Financial Data and Financial Econometrics

• Ordinary Least Squares Regression Analysis - What is OLS?

• Ordinary Least Squares Regression Analysis - Testing Hypotheses

• Ordinary Least Squares Regression Analysis - Diagnostic Testing

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Econometrics

• Literally means “measurement in economics”

• More practically it means “the application of statistical techniques to problems in economics”

• In this course we focus on problems in financial economics

• Usually, we will be trying to explain the behavior of a financial variable

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Econometric Model Building

Assess implications for theory

In terpret m odel

Satisfactory

R e-estim ate m odel using better techniquesC ollect better dataR eform ulate m odel

U nsatisfactory

5. Evaluate estim ation results

4. Estim ate m odel

3. C ollect data

2. D erive estim able m odel

1. U nderstand finance theory

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Financial Data

• What sorts of financial variables do we usually want to explain?– Prices - stock prices, stock indices, exchange rates– Returns - stock returns, index returns, interest rates– Volatility– Trading volumes– Corporate finance variables

• Debt issuance, use of hedging instruments

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Time Series Data

• Time-series data are data arranged chronologically, usually at regular intervals– Examples of Problems that Could be Tackled Using a

Time Series Regression• How the value of a country’s stock index has varied with

that country’s macroeconomic fundamentals.

• How a company’s stock returns has varied when it announced the value of its dividend payment.

• The effect on a country’s currency of an increase in its interest rate

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Cross Sectional Data

• Cross-sectional data are data on one or more variables collected at a single point in time

• e.g. A sample of bond credit ratings for UK banks

– Examples of Problems that Could be Tackled Using a Cross-Sectional Regression

• The relationship between company size and the return to investing in its shares

• The relationship between a country’s GDP level and the probability that the government will default on its sovereign debt.

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Panel Data

• Panel Data has the dimensions of both time series and cross-sections

• e.g. the daily prices of a number of blue chip stocks over two years.

– It is common to denote each observation by the letter t and the total number of observations by T for time series data,

– and to to denote each observation by the letter i and the total number of observations by N for cross-sectional data.

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Econometrics versus Financial Econometrics

– Little difference between econometrics and financial econometrics beyond emphasis

– Data samples• Economics-based econometrics often suffers from paucity of

data• Financial economics often suffers from infoglut and signal to

noise problems even in short data samples

– Time scales• Economic data releases often regular calendar events• Financial data are likely to be real-time or tick-by-tick

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Economic Data versus Financial Data

• Financial data have some defining characteristics that shape the econometric approaches that can be applied– outliers– trends– mean-reversion– volatility clustering

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Outliers

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Trends

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Mean-Reversion (with Outliers)

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More Mean-Reversion

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Volatility Clustering

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Basic Data Analysis

• All pieces of empirical work should begin with some basic data analysis– Eyeball the data– Summarize the properties of the data series– Examine the relationship between data series

• Most powerful analytic tools are your eyes and your common sense– Computers still suffer from “Garbage in - garbage

out”

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Basic Data Analysis

• Eyeballing the data helps establish presence of – trends versus mean reversion– volatility clusters– key observations

• outliers – data errors?

• turning points

• regime changes

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Basic Data Analysis

• Summary statistics– Average level of variable

• Mean, median, mode

– Variability around this central tendency• Standard deviations, variances, maxima/minima

– Distribution of data• Skewness, kurtosis

– Number of observations, number of missing observations

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Basic Data Analysis

• Since we are usually concerned with explaining one variable using another– “trading volume depends positively on volatility”

• relationships between variables are important– cross-plots, multiple time-series plots– correlations (covariances)– multi-collinearity

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Basic Data Manipulations

• Taking natural logarithms

• Calculating returns

• Seasonally adjusting

• De-meaning

• De-trending

• Lagging and leading

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The basic story

• y is a function of x

• y depends on x

• y is determined by x

“the spot exchange rate depends on relative price levels and interest rates…”

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Terminology

• y is the x’s are the – predictand predictors– regressand regressors– explained variable explanatory variables– dependent variable independent variables– endogenous variable exogenous variables– left hand side variable right hand side variables

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Data

• Suppose we have n observations on y and x:

cross section

yi = α + β xi + ui i = 1, 2, …, n

time series

yt = α + β xt + ut t = 1, 2, …, n

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Errors

• Where does the error come from?– Randomness of (human) nature

• men and markets are not machines

– Omitted variables• men and markets are more complex than the models

we use to describe them. Everything else is captured by the error term

– Measurement error in y• unlikely in financial applications

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Objectives

• to get good point estimates of α and β given the data

• to understand how confident we should be in those estimates

• both will allow us to make statistical inferences on the true form of the relationship between y and x (“test the theory”)

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Simple Regression: An Example• We have the following data on the excess returns on a

fund manager’s portfolio (“fund XXX”) together with the excess returns on a market index:

• We want to find whether there is a relationship between x and y given the data that we have. The first stage would be to form a scatter plot of the two variables.

Year, t Excess return= rXXX,t – rft

Excess return on market index= rmt - rft

1 17.8 13.72 39.0 23.23 12.8 6.94 24.2 16.85 17.2 12.3

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Graph (Scatter Diagram)

0

5

10

15

20

25

30

35

40

45

0 5 10 15 20 25

Excess return on market portfolio

Ex

cess

re

turn

on

fu

nd

XX

X

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Finding the Line of Best Fit

• We can use the general equation for a straight line,

y = α + βx

to get the line that best “fits” the data. • But this equation (y = α + βx) is completely

deterministic. • Is this realistic? No. So what we do is to add a random

disturbance term, u into the equation.

yt = + xt + ut

where t = 1, 2, 3, 4, 5

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Determining the Regression Coefficients

• So how do we determine what and are? • Choose and so that the distances from the data

points to the fitted lines are minimised (so that the line fits the data as closely as possible)

• The most common method used to fit a line to the data is known as OLS (ordinary least squares).

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Ordinary Least Squares

• What we actually do is

1. take each vertical distance between the data point and the fitted line

2. square it and

3. minimize the total sum of the squares (hence least squares).

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y = - 1.7366 + 1.6417x

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10

15

20

25

30

35

40

5 10 15 20 25

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Algebra Alert!!!!!

• Tightening up the notation, let

• yt denote the actual data point t

• denote the fitted value from the regression line

• denote the residual, yt -

ty

tytu

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How OLS Works

• So min. , or minimise . This is known as the residual sum of squares.

• But what was ? It was the difference between the actual point and the line, yt - .

• So minimizing is equivalent to minimizing

with respect to and .

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24

23

22

21 ˆˆˆˆˆ uuuuu

ty

tu

5

1

2ˆt

tu

2ˆ tt yy 2ˆtu

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Coefficient Estimates

of valueestimated e thˆˆ

of valueestimated e thˆ

:estimtes OLS gives

zero toequal setting and ˆ and ˆ wrt RSS atingdifferenti

ˆˆˆ

222

1 1

22

xy

SS

yxTyxyyxxS

xTxxxS

xyyyRSS

xx

xy

ttttxy

ttxx

T

t

T

ttttt

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What do we Use and For?

• In the CAPM example used above, optimising would lead to the estimates • = -1.74 and • = 1.64.

• We would write the fitted line as:

tt xy 64.174.1ˆ

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• If an analyst tells you that she expects the market to yield a return 20% higher than the risk-free rate next year, what would you expect the return on fund XXX to be?

• Solution: We can say that the expected value of y = “-1.74 + 1.64 * value of x”, so plug x = 20 into the equation to get the expected value for y:

What do we Use and For?

06.312064.174.1ˆ iy

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Is Using OLS a Good Idea?

• Yes, since given some assumptions (see later) least squares is BLUE– best, linear, unbiased estimator

• OLS is consistent– as sample size increases, estimated coefficients tend

towards true values

• OLS is unbiased– Even in small samples, estimated coefficients are on

average equal to true values

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Is Using OLS a Good Idea? (cont.)

• OLS is efficient– no other linear estimator has a smaller variance

around the estimated coefficient values– some non-linear estimators may be more

efficient

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Testing Hypotheses

• Once you have regression estimates (assuming the regression is a “good” one) you take the results to the theory:“Theory says that the intercept should be zero”

“Theory says that the coefficient on prices should be unity”

“Theory says that the coefficient on domestic money should be unity and the coefficient on foreign money should be minus unity”

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Testing Hypotheses (cont.)

• Testing these statements is called hypothesis testing

• This involves comparing the estimated coefficients with what theory suggests

• In order to say whether the estimates are “too far” from theory we need some measure of the precision of the estimated coefficients

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Standard Errors

• Based on a sample of data, you have estimated the coefficients and

• How much are these estimates likely to alter if different samples are chosen?

• The usual measure of this degree of uncertainty is the standard error of the coefficient estimates

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Standard Errors (cont.)

• Algebraically, given some crucial assumptions, standard errors can be computed as follows:

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22

2

1ˆ

ˆ

xTxsSE

xTxT

xsSE

t

t

t

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Error Variance

• σ2 is the variance of the error or disturbance term, u

• this is unobservable

• we approximate it with the variance of the residual terms, s2

2

ˆ

2

ˆ

2

ˆˆ2

T

us

T

u

T

uus

t

ttt

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Standard Errors

• SE are smaller as – T increases,

• more data makes precision of estimated coefficients higher

– the variance of x increases,• more dispersion of dependent variable about its mean,

makes estimated coefficients more precise

– s decreases• better the fit of the regression (smaller residuals), the more

precise are estimates

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Null and Alternative Hypotheses

• So now you have the coefficient estimates and the associated standard errors.

• You now want to test the theory.

Five-Step Process:

Step 1: Draw up the null hypothesis (H0)Step 2: Draw up the alternative hypothesis

(H1 or HA)

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Null Hypothesis• Usually, the null hypothesis is what theory suggests:

e.g. testing the ability of fund mangers to outperform the index

• EMH suggests αj = 0,

• so, H0: αj = 0 (fund managers earn zero risk adjusted excess returns)

jtftmtjjftjt uRRRR

returnadjustedrisk expected

tat time j fundofreturn excess

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Alternative Hypothesis

• The alternative is more tricky

• Usually the alternative is just that the null is wrong:– H1: α 0 (fund managers earn non-zero risk

adjusted excess returns; fund managers underperform or out-perform)

• But sometimes is more specific– H1: α < 0 (fund managers underperform)

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Confidence Intervals

• Suppose our point estimate for α is 0.058 for fund XXXX and the associated standard error is 0.025 based on 20 observations

• Has fund XXXX outperformed?• Can we be confident that the true α is

different to zero?

Step 3: Choose your level of confidence

Step 4: Calculate confidence interval

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Confidence Interval (cont.)

• Convention is to use 95% confidence levels

• Confidence interval is then

– tcritical is appropriate percentile (eg 97.5th) of the t-distribution with T-2 degrees of freedom

• 97.5th percentile since two-sided test

• 2 degrees of freedom were lost in estimating 2 coefficients

ˆˆ,ˆˆ SEtSEt criticalcritical

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Confidence Interval (cont.)

• We are now 95% confident that the true value of alpha lies between

0.058 - 2.1009*0.025 = 0.0059 and

0.058 + 2.1009*0.025 = 0.1105

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Making inferences

Step 5: Does the value under the null hypothesis lie within this interval?

• No (null was that alpha = 0)– So we can reject the null hypothesis that fund

XXXX earns a zero risk adjusted excess return– and accept the alternative hypothesis– we reject the restriction implied by theory

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Making inferences (cont.)

• Suppose our standard error was 0.03

• The confidence interval would have been -0.005 to 0.121

• The value under the null is within this range– We cannot reject the null hypothesis that fund

XXX only earns a zero risk adjusted return– NOTE we never accept the null - hypothesis

testing is based on the doctrine of falsification

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Significance Tests

• Instead of calculating confidence intervals we could calculate a significance test

Step 4: Calculate test statistic

α* is value under the null

Step 5: Compare test statistic to critical value, tcritical

32.2025.0

0.0058.0ˆ

ˆ *

SE

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Significance Tests (cont.)

Step 6: Is test statistic in the non-rejection or acceptance region?

2.5% rejectionregion

2.5% rejectionregion

95% acceptance region

+2.1009-2.1009

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One-Sided tests (cont.)

• Suppose the alternative hypothesis was– H1: α < 0

• We then perform one-sided tests– Confidence interval is

– Significance test statistic is compared to tcritical

– tcritical is based on 95th percentile (not 97.5th)

ˆˆ, SEtcritical

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Type I and Type II Errors

• Where did 95% level of confidence come from?– Convention

• What does it mean?– We are going to reject the null when it is

actually true 5% of the time– This is a Type I error

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Type I and Type II Errors (cont.)

– Type II error is when we fail to reject the null when it was actually false

– To reduce Type I errors, we could use 99% confidence level

• this would widen our CI, raise the critical value

• making it less likely to reject the null by mistake

• but also making it less likely we correctly reject the null

• so raises Type II errors

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Which is Worse?

• This depends on the circumstances– In Texas, the null hypothesis is that you are

innocent and if the null is rejected you are killed

• Type I errors are very important (to the accused)

– But if tests are “weak” there is low power to discriminate and econometrics cannot inform theory

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Statistical Significance I

• Can we say with any degree of certainty that the true coefficient is statistically different from zero?

• t-statistic and P-value– t-stat is the coefficient estimate/its standard error– rule-of-thumb is that |t-stat|>2 means we can be 95%

confident that the true coefficient is not equal to zero– P-value gives probability that true coefficient is zero given

the estimated coefficient and its standard error

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 2.02593034 0.127404709 15.90153 4.835E-12 1.7582628 2.29359791X Variable 1 0.48826704 0.044344693 11.01072 1.99E-09 0.3951022 0.58143186

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Statistical Significance II

• Can we say with any degree of certainty that the true coefficient is statistically different from zero?

• Confidence intervals– Give range within which we can be 95% confident that

the true coefficient lies • actual coefficients are 2.0 and 0.5

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 2.02593034 0.127404709 15.90153 4.8355E-12 1.7582628 2.29359791X Variable 1 0.48826704 0.044344693 11.01072 1.99E-09 0.3951022 0.58143186

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Statistical Inference II (cont.)

• t-test coeff = 0.5 is (0.488 - 0.5)/0.044 = -0.26• t-test coeff = 0.6 is (0.488 - 0.6)/0.044 = -2.52• critical value of t-test (17 d.f.) is 2.11

– cannot reject null that true coefficient is 0.5

– can reject null that true coefficient is 0.6 in favour of alternative that true coefficient is different to 0.6

• with 95% confidence

• or at the 5% level of significance

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Economic Significance

• As x increases by 1 unit, y on average increases by 0.488 units

• The econometrician has to decide whether this is economically important– Depends on magnitudes of x and y and the variability of x

and y

– Very important in finance to check economic importance of results

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 2.02593034 0.127404709 15.90153 4.8355E-12 1.7582628 2.29359791X Variable 1 0.48826704 0.044344693 11.01072 1.99E-09 0.3951022 0.58143186

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Some Real Data

• annually from 1800+ to 2001

• spot cable [dollars per pound] (spot)

• consumer price indices (ukp, usp)

• long-term interest rates (uki, usi)

• stock indices (ukeq, useq)

• natural log of each series (l...)

• log differences of each series (d…)

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Excel Regression Output

Regression StatisticsMultiple R 0.245486731R Square 0.060263735Adjusted R Square 0.038906093Standard Error 0.092479061Observations 181

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept -0.00790941 0.007344336 -1.07694 0.2829809 -0.02240372 0.00658489dukp -0.28954448 0.11451444 -2.52845 0.01233588 -0.51554277 -0.06354619dusp 0.393294362 0.140052233 2.808198 0.00554476 0.116896338 0.66969239duki -0.06224627 0.083163402 -0.74848 0.45516881 -0.22637218 0.10187964dusi -0.02335464 0.069677157 -0.33518 0.73788573 -0.16086497 0.11415569

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Assumptions of OLS

• OLS is BLUE if the error term, u, has:– zero mean: E(ui) = 0 all i

– common variance: var(ui)=σ2 all i

– independence: ui and uj are independent (uncorrelated) for all i j

– normal: ui are normally distributed for all i

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Problems with OLS

• What the problem means

• How to detect it

• What it does to our estimates and inference

• How to correct for it

• Key Problems: Multi-Collinearity, Non-Normality, Heteroskedasticity, Serial Correlation.

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Multi-collinearity

• What it means– Regressors are highly intercorrelated

• How to detect it– If economics of model is good, high R-squared, lots of

individually insignificant (t-stats) but jointly significant (F-tests) regressors. Also, via high VIF values > 10.

• What it does– Inference is hard because std errors are blown up

• How to correct for it– Get more data; sequentially drop regressors.

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Non-Normality

• What it means– Over and above assumptions about mean and variance

of regression errors, OLS also assumes they are normally distributed

• How to detect non-normality– If normal, skew = 0, kurtosis = 3

– Jarque-Bera test• J-B = n[S2/6 + (K-3)2/24]

• distributed χ2(2) so CV is approx 6.0– J-B>6.0 => non-normality

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Non-Normality (cont.)

• What it does (loosely speaking)• skewness means coefficient estimates are biased.• excess kurtosis means standard errors are understated.

• How to correct for it– skewness can be reduced by transforming the data

• take natural logs• look at outliers

– kurtosis can be accounted for by adjusting the degrees of freedom used in standard tests of coefficient on x

• use k(T-2) d.f. instead of (T-2)

• 1/k = 1+[(Ku - 3)(Kx - 3)]/2T

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Heteroskedasticity

• What it means– OLS assumes common variance or homoscedasticity

• var(ui) = σ2 for all i

– Heteroscedasticity is when the variance varies• often variance gets larger for larger values of x

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Detecting Heteroskedasticity

– Plot residuals as time series or against x

– White test• Regress squared residuals on x’s, x2’s and cross products.

– Reset test• Regress residuals on fitted y2, y3, etc. Significance indicates

heteroscedasticity

– Goldfeld-Quandt test• Split sample into large x’s and small x’s, fit separate

regressions and test for equality of error variances

– Breusch-Pagan test• σ2 = a + bx + cz … so test b = c = 0

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White Test

White Heteroskedasticity Test:

F-statistic 1.497731 Probability 0.192863Obs*R-squared 7.427565 Probability 0.190734

Dependent Variable: RESID^2Variable Coefficient Std. Error t-Statistic Prob.

C 0.006270 0.002497 2.511094 0.0129DUKP -0.001079 0.034449 -0.031327 0.9750

DUKP^2 -0.196008 0.282897 -0.692862 0.4893DUKP*DUSP 0.033795 0.614692 0.054979 0.9562

DUSP 0.034454 0.045411 0.758711 0.4490DUSP^2 0.694162 0.398780 1.740713 0.0835

R-squared 0.041036 Mean dependent var 0.008361Log likelihood 397.3308 F-statistic 1.497731Durbin-Watson stat 1.774378 Prob(F-statistic) 0.192863

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Implications of Heteroskedasticity

• OLS coefficient estimates are unbiased• OLS is inefficient

– has higher variance than it should

• OLS estimated standard errors are biased– if σ2 is positively correlated with x2 (usual case) then

estimated standard errors are too small

– so inference is wrong• we become too confident in our estimated coefficients

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Correcting for Heteroskedasticity

• If we know the nature of the heteroskedasticity it is best to take this into account in the estimation– use weighted least squares

– “deflate” the variable by the appropriate measure of “size”

• Usually, we don’t know the functional form– so correct the standard errors so that inference is valid

– White standard errors alter the OLS std errors and asymptotically give reasonable inference properties

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White Standard Errors

Dependent Variable: DSPOTMethod: Least SquaresDate: 07/17/03 Time: 21:41Sample(adjusted): 1821 2001Included observations: 181 after adjusting endpointsWhite Heteroskedasticity-Consistent Standard Errors & Covariance

Variable Coefficient Std. Error t-Statistic Prob.

C -0.007288 0.006605 -1.103411 0.2713DUKP -0.310380 0.102816 -3.018788 0.0029DUSP 0.379539 0.197376 1.922921 0.0561

R-squared 0.055141 Mean dependent var -0.006391Adjusted R-squared 0.044525 S.D. dependent var 0.094332S.E. of regression 0.092208 Akaike info criterion -1.913097Sum squared resid 1.513423 Schwarz criterion -1.860083Log likelihood 176.1352 F-statistic 5.193965Durbin-Watson stat 2.115206 Prob(F-statistic) 0.006422

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Serial Correlation

• OLS assumes no serial correlation– ui and uj are independent for all i j

• In cross-section analysis, residuals are likely to be correlated across individuals– e.g. common shocks

• In time series analysis, today’s error is likely to be related to (correlated with) yesterday’s residual– autocorrelation or serial correlation

– maybe due to autocorrelation in omitted variables

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Detecting Serial Correlation

• Durbin-Watson test statistic, d– assumes errors ut and ut-1 have (positive) correlation p

– tests for significance of p on basis of correlation between residuals u^t and u^t-1

– only valid in large samples,

– only tests first order correlation

– only valid if there are no lagged dependent variables (yt-i) in regression

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Detecting Serial Correlation (cont.)

• d lies between 0 and 4– d = 2 implies residuals uncorrelated.

• D-W provide upper and lower bounds for d– if d < dL then reject null of no serial correlation

– if d > dU then reject null hypothesis of no serial correlation

– if dL< d < dU then test is inconclusive.

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Durbin-Watson

Dependent Variable: DSPOTMethod: Least SquaresDate: 07/17/03 Time: 21:41Sample(adjusted): 1821 2001Included observations: 181 after adjusting endpointsWhite Heteroskedasticity-Consistent Standard Errors & Covariance

Variable Coefficient Std. Error t-Statistic Prob.

C -0.007288 0.006605 -1.103411 0.2713DUKP -0.310380 0.102816 -3.018788 0.0029DUSP 0.379539 0.197376 1.922921 0.0561

R-squared 0.055141 Mean dependent var -0.006391Adjusted R-squared 0.044525 S.D. dependent var 0.094332S.E. of regression 0.092208 Akaike info criterion -1.913097Sum squared resid 1.513423 Schwarz criterion -1.860083Log likelihood 176.1352 F-statistic 5.193965Durbin-Watson stat 2.115206 Prob(F-statistic) 0.006422

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Implications of Serial Correlation

• With no lagged dependent variable (so d is a valid test)– OLS coefficient estimates are unbiased

– but inefficient

– estimated standard errors are biased• so inference is again wrong

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Correcting for 1st Order Serial Correlation

• Rule of thumb:– if d < R2 then estimate model in first difference form

yt = α + β xt + ut

yt-1 = α + β xt-1 + ut-1

yt - yt-1 = β( xt - xt-1) + (ui - ut-1)

– so we can recover the regression coefficients (but not the intercept).

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Implications of Serial Correlation

• With lagged dependent variables in regression (when DW test is invalid).

• OLS coefficient estimates are inconsistent– even as sample size increases, estimated coefficient

does not converge on the true coefficient (i.e. it is biased)

• So inference is wrong.

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Using a Dummy Variable to Test Changes in Slope & Intercept

• What if and change over time? So, what we do is add two additional RHS variables:

yt = + xt + Dt + (Dt xt) + ut

where, t = 1, 2, 3, … T.

Dt = 1 for 2003 - 2004 period, 0 otherwise.

= measures change in intercept during 2003 - 2004.

= measures change in slope during 2003 - 2004.