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Lecture 3.2 on Model Specification: Specification

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Econometrics: Methods and Applications

Erasmus University Rotterdam

4.6 (756 ratings) | 99K Students Enrolled

Welcome! Do you wish to know how to analyze and solve business and economic questions with data analysis tools? Then Econometrics by Erasmus University Rotterdam is the right course for you, as you learn how to translate data into models to make forecasts and to support decision making. * What do I learn? When you know econometrics, you are able to translate data into models to make forecasts and to support decision making in a wide variety of fields, ranging from macroeconomics to finance and marketing. Our course starts with introductory lectures on simple and multiple regression, followed by topics of special interest to deal with model specification, endogenous variables, binary choice data, and time series data. You learn these key topics in econometrics by watching the videos with in-video quizzes and by making post-video training exercises. * Do I need prior knowledge? The course is suitable for (advanced undergraduate) students in economics, finance, business, engineering, and data analysis, as well as for those who work in these fields. The course requires some basics of matrices, probability, and statistics, which are reviewed in the Building Blocks module. * What literature can I consult to support my studies? You can follow the MOOC without studying additional sources. Further reading of the discussed topics (including the Building Blocks) is provided in the textbook that we wrote and on which the MOOC is based: Econometric Methods with Applications in Business and Economics, Oxford University Press. The connection between the MOOC modules and the book chapters is shown in the Course Guide – Further Information – How can I continue my studies. * Will there be teaching assistants active to guide me through the course? Staff and PhD students of our Econometric Institute will provide guidance in January and February of each year. In other periods, we provide only elementary guidance. We always advise you to connect with fellow learners of this course to discuss topics and exercises. * How will I get a certificate? To gain the certificate of this course, you are asked to make six Test Exercises (one per module) and a Case Project. Further, you perform peer-reviewing activities of the work of three of your fellow learners of this MOOC. You gain the certificate if you pass all seven assignments. Have a nice journey into the world of Econometrics! The Econometrics team

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Skills You'll Learn

Linear Regression, Time Series, Econometrics, Regression Analysis

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PK

Dec 12, 2016

This is a fantastic MOOC: it has depth, exercise questions with solutions, challenging assignments and background material. The quality of the lecture videos is excellent!

GT

Oct 23, 2018

Rigorous condensed econometrics course with clear instruction. Great for my review of the subject, and most likely for anyone new to it who has the prerequisite skills.

Aus der Unterrichtseinheit

Model Specification

Lecture 3.2 on Model Specification: Specification9:04

Unterrichtet von

Philip Hans Franses
Prof. Dr.
Christiaan Heij
Dr.
Michel van der Wel
Dr.
Dennis Fok
Prof. Dr.
Richard Paap
Prof. Dr.
Dick van Dijk
Prof. Dr.
Erik Kole
Dr.
Francine Gresnigt
PhD candidate
Myrthe van Dieijen
PhD candidate

Skript

Welcome.

In this lecture you will learn how to specify and

select a good econometric model.

We start with the familiar model where the dependent variable y is explained by a set

of variables collected in X.

Here, y can be a stock index return and

X a number of variables that may explain movements in the stock index.

We write this relationship either explicitly for

each individual observation as done on the top of the page here, or

in matrix form, as was discussed in lecture two.

The question we'll address now is which variables to include in the matrix X.

It turns out that there's a tough trade off that we face.

If one considers a model with a small number of variables,

there is the risk that relevant variables are missed, and

thus actually too few variables are included.

This will lead to an estimation bias.

If one, however,

considers a model with too many variables, there's an efficiency loss.

Now let us develop the results on a previous slide.

We compare two models.

Suppose that the data-generating process, DGP in short,

contains two group of explanatory variables, X1 and X2.

We contrast this with the actual estimated model which only contains X1.

In this model we denote the estimator of beta1 by b(r),

where r stands for restricted

as we've restricted beta2 to zero.

Also a tilde is added to the disturbance term,

to indicate that it is different from the one in the DGP.

The estimators of beta1 and beta2 in the DGP are then ordered by b(1) and b(2).

Now I invite you to answer the following test question

to relate the expected value of b(r) to the values beta1 and beta2.

The answer uses the famous OLS formula for the regression of y on X1 alone.

Then we plug in the expression for y.

Here it is key to use the equation that generated the data, so

with both X1 and X2.

Then we split this into three parts.

The first part becomes beta1,

the second part a certain matrix times beta2 and the last part is zero.

This gives us the first result.

The restricted estimator will be biased unless beta2 is zero, or

X1 and X2 are completely orthogonal, such that the product and thus P is zero.

We refer to this bias as the omitted variable bias.

Now we turn to the efficiency part.

Efficiency concerns the variance of our estimators.

We prefer estimators that have no or small bias with low variance.

An estimator with the lowest possible variance is called efficient.

The variance of the restricted estimator, b(r), is equal to the variance of

the unrestricted estimator, b(1) minus a positive semi-definite term,

such that the variance of b(1) is always larger than that of b(r).

This result will be considered further in this lecture's training exercise.

While the benefit of adding variables is bias reduction,

a cost is thus increased variance.

The next step is to translate this finding into some measures, or

Metrics, that we can use to find a good trade off between bias and efficiency.

We turn to two commonly used decision metrics, information criteria and

out-of-sample prediction.

Often there's a preference for

small models in the sense that a limited number of variables are included.

When adding variables, at a certain stage the added benefit of yet

another variable will be relatively small,

and it is good to stop adding variables to the model.

Information criteria capture this idea.

They study the goodness of fit of a model, here captured

with the standard error of the regression as defined in lecture two,

but impose a penalty on the number of parameters k.

Two commonly used information criteria are the Akaike information criterion,

abbreviated with AIC, and

the Bayesian information criterion, abbreviated with BIC.

For both the AIC and

BIC the value is equal to the log of the squared standard error of the regression

plus a term that is a function of k, the number of variables in the model.

The two information criteria differ in the penalty they impose on the number of

parameters.

When comparing models, a lower value of the information criteria is preferred

as we aim for a low standard error of the regression.

Now I invite you to go examine which information criterion

imposes the strongest penalty on the number of variables.

The penalty on the number of parameters k is 2/n for

the AIC and this is log(n) over n for BIC.

Thus which criterion imposes the strongest penalty

depends on the number of observations n.

When log(n) is larger than 2, the BIC imposes a stronger penalty.

Thus when n is larger than the square of e,

the BIC imposes a greater penalty from which we conclude that for eight or

more observations, BIC imposes a stronger penalty than AIC.

The information criteria are based on

so-called in-sample results: using all observations in a sample.

Often we're also interested in the predictive performance of our model.

This can be in a time series sense, that we want to forecast a stock

price to earn some money, but also if you have data on household consumption and

want to predict whether they will buy a certain product or not.

In such cases, the full sample can be split in an in-sample part,

often referred to as the training sample, and an out-of-sample part.

The observations in the second out of sample part

are kept out of the main analysis, for example when estimating beta,

and they're only used to examine the predictive ability of the model.

Two commonly used out of sample criteria are the root mean squared error, RMSE, and

the mean absolute error, MAE.

Both criteria consider the difference between the actual observation y(i) and

the predicted value, y-hat(i),

but they differ slightly in how the prediction errors are averaged.

In both cases, a lower value means a better model.

Now let us return to the problem that a researcher faces,

how to decide which variables to include in X.

If you consider removing a group of regressors, you can use an F-test for

a joint significance of the second group of coefficients, or simply a t-test

if you wish to remove only a single variable.

However, be aware that these tests are only concerned with the significance and

do not incorporate the bias efficiency trade off.

If you already have a set of candidate models that differ in

the number of parameters, information criteria can be of use.

These take into account that small models are preferred

if more complex models do not perform sufficiently better.

Here you can also consider using out-of-sample prediction.

If the goal is prediction and there are a number of candidate models,

you may as well pick the one that has the most predictive power,

and provides the lowest root mean squared error or mean absolute error.

Very often we're, however, not fortunate enough to start

with two groups of regressors, X1 and X2, or with a candidate set of models,

and we need to get just one model first.

In this case, iterative selection methods can be of great help.

These come in two variants:

general to specific and specific to general.

In the first you start with the most general model,

including as many variables as are at hand.

Then check whether one or more variables can be removed from the model.

This can be based on individual t-tests,

or a joint F-test in case of multiple variables.

In case you remove one variable at a time,

the variable with the lowest absolute t-value is removed from the model.

The model is estimated again without that variable, and the procedure is repeated.

The procedure continues until all remaining variables are significant.

The specific to general approach follows the same logic, but

starts with a very small model, sometimes even only consisting of the constant term.

Variables get added one at a time,

choosing the one that has the largest absolute t-statistic.

This procedure is repeated until no significant variables can

be added anymore.

Both procedures have pros and cons.

The specific-to-general approach starts small, which is appealing.

However, many variations need to be tried at the initial steps.

Also, it can easily happen that important variables are missing in initial phases so

that initial tests are performed in mis-specified models.

Now I invite you to make the training exercise,

to train yourself with the topics of this lecture.

You can find this exercise on the web site.

And this concludes our lecture on specification.

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