ECONOMETRICS FOR FINANCE - a lecture

ECONOMETRICS FOR FINANCE

The Nature and Purpose of Econometrics

•          Financial econometrics:

                The application of statistical and mathematical techniques to problems in finance.

EXAMPLES OF THE FINANCIAL PROBLEMS

1. Testing whether financial markets are informational efficient.
2. Testing whether the CAPM or APT represent superior models for the determination of returns on risky assets.
3. Measuring and forecasting the volatility of stock returns.
4. Modelling long-term relationships between prices and exchange rates
5. Testing technical trading to determine which makes the most money.
6. Testing the hypothesis that earnings or dividend announcements have any effect on stock prices.
7. Testing whether markets react to news.
8. Forecasting the correlation between the returns to the stock indices.

TYPES OF DATA AND NOTATION

•          There are 3 types of data which econometricians might use for analysis:

                1. Time series data

                2. Cross-sectional data

                3. Panel data, a combination of 1. & 2.

•          The data may be quantitative (e.g. exchange rates, stock prices, interest rates), or qualitative (e.g. day of the week).

•          Examples of time series data

                Series                                                    Frequency

                GNP or unemployment                 monthly, quarterly or annually

                Budget deficit                               annually

                Money supply                              weekly, monthly

                Stock market index                      daily, weekly, monthly

Time Series versus Cross-sectional Data

•          Time Series Data

                - The variation in the value of a country’s stock index with that of economic fundamentals.

                - The variation in the value of a company’s stock price with the announced value of its dividend                                payment.

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

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

                - A survey of usage of internet stock  broking services

                - A survey of usage of internet banking

                - Investors behaviour

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

Cardinal, Ordinal and Nominal Numbers

•          Another way in which we could classify numbers is according to whether they are cardinal, ordinal, or nominal.

•          Cardinal numbers are those where the actual numerical values that a particular variable takes have meaning, and where there is an equal distance between the numerical values.

–        Examples of cardinal numbers would be the price of a share or the number of houses in a street.

•          Ordinal numbers can only be interpreted as providing a position or an ordering.

–        Examples of ordinal numbers would be the position of a runner in a race or stock or bond rating.

•          Nominal numbers occur where there is no natural ordering of the values at all.

–        Such data often arise when numerical values are arbitrarily assigned..

•          Cardinal, ordinal and nominal variables may require different modeling approaches or at least different treatments.

Returns in Financial Modelling

•          It is preferable not to work directly with asset prices, so we usually convert the raw prices into a series of returns. There are two ways to do this:

•          We normally ignore any dividend payments, or alternatively assume that the price series have been already adjusted to account for them.

A Disadvantage of using Log Returns

•          There is a disadvantage of using the log-returns. The simple return on a portfolio of assets is a weighted average of the simple returns on the individual assets.

•          But this does not work for the continuously compounded returns.

Some Points to consider when reading papers in the finance literature

1. Does the paper involve the development of a theoretical model or is it merely a technique looking for an application, or an exercise in data mining?

2. Is the data of “good quality”? Is it from a reliable source? Is the size of the sample sufficiently large for asymptotic theory to be invoked?

3. Have the techniques been validly applied? Have diagnostic tests for violations been conducted for any assumptions made in the estimation of the model?

4. Have the results been interpreted sensibly? Is the strength of the results exaggerated? Do the results actually address the questions posed by the authors?

5. Are the conclusions drawn appropriate given the results, or has the importance of the results of the paper been overstated?

TIME SERIES DATA ISSUES

•          Heteroscedasticity

•          Auto correlation

•          Multi colinearity

Heteroscedasticity

•          If the errors do not have a constant variance, we can say that they are heteroscedastic. It implies that values in a series have different numbers and dimensions. 

•          White’s general test for heteroscedasticity is one of the best approaches because it makes few assumptions about the form of the heteroscedasticity.

•          Transforming the variables into logs or reducing by some other measure of “size”.

•          The concept of a lagged value

Consequences of Using OLS in the Presence of Heteroscedasticity

•          OLS estimation still may gives biased coefficient estimates.

•          This implies that if we still use OLS in  the presence of heteroscedasticity, our standard errors could be inappropriate and hence any inferences we make could be misleading.

•          Whether the standard errors calculated using the usual formulae are too big or too small will depend upon the form of the heteroscedasticity.

•          If the form (i.e. the cause) of the heteroscedasticity is known, then we can use an estimation method which takes this into account (called generalised least squares, GLS or ARCH/GARCH).

Autocorrelation

•          If there are patterns in the residuals from a model, we say that they are autocorrelated or values in a series are following a pattern.

1. Positive Autocorrelation
2. Negative Autocorrelation
3. No pattern in residuals – No autocorrelation

Detecting Autocorrelation

The Durbin-Watson (DW) is a test for first order autocorrelation - i.e. it assumes that the relationship is between an error and the previous one.

                Unfortunately, DW has 2 critical values, an upper critical value (d_u) and a lower critical value (d_L), and there is also an intermediate region where we can neither reject nor not reject H₀.

The Durbin-Watson Test: Interpreting the Results

Conditions which Must be Fulfilled for DW to be a Valid Test

                1. Constant term in regression

                2. Regressors are non-stochastic

                3. No lags of dependent variable

Consequences of Ignoring Autocorrelation

•          The coefficient estimates derived using OLS are still unbiased, but they are inefficient, i.e. even in large sample sizes.

•          Thus, if the standard error estimates are inappropriate, there exists the possibility that we could make the wrong inferences.

•          R² is likely to be inflated relative to its “correct” value for positively correlated residuals.

Multicollinearity

•          This problem occurs when the explanatory variables are very highly correlated with each other.

•          Problems if multicollinearity is present but ignored

                - R² will be high but the individual coefficients will have high standard errors.

                - The regression becomes very sensitive to small changes in the specification.

                - Thus confidence intervals for the parameters will be very wide, and significance tests might     therefore give inappropriate conclusions.

•          The easiest way to measure the extent of multicollinearity is simply to look at the matrix of correlations between the individual variables.

Solutions to the Problem of Multicollinearity

•          “Traditional” approaches, such as principal components. But these usually bring more problems than they solve.

•          Some econometricians argue that if the model is otherwise OK, just ignore it.

•          The easiest ways to “cure” the problems are

                - drop one of the collinear variables

                - transform the highly correlated variables into a ratio

                - go out and collect more data e.g.

                                                - a longer run of data

                                                - switch to a higher frequency

Presented by Dr. Babar Zaheer Butt to the MS/PhD Students at Iqra University Islamabad 

Download