Microeconometria

L. Cembalo

Chow Test for structural change

Question to answer is: Immagine we know there has been something that made a functional relationship changing over a specific event. How do we test if data carries that change?

We may have two different effects: on the intercept and/or on the slope coefficient

example: data on disposable income and personal savings in billion dollars for USA, period 1970-1995. In 1982 USA faced the worst peacetime recession

Chow TestModello: Savings ; Income

Periodo: 1970-1995

Potenziale break strutturale nel 1982

Periodo1970-1981 Periodo1982-1995 Periodo1970-1995

n1=12 n2=14 n0(n1+n2)=26

Last equation assumes that there is no difference between the two time

F-Test su RSS

F=(RSSr - RSSur)/k/((RSSur)/(n1+n2-2k))

this is because RSS1 and RSS2 are deemed independent, so that RSSur=RSS1+RSS2

F(k, n1+n2-2k) gradi di libertà

Stata for numerical example

F-Test su RSS - caveats

All this works if the error variances in the regressions (the two unrestricted) are the same. It should be tested

The Chow test will tell us only if a structural change occurs with our saying whether it depends on slope and/or intercept term/s (later with dummies we will solve this point)

Chow test assumes we know where the break occurred

Dummy variables effetto sull’intercetta

Modello effetti su y= mesi per trovare lavoro; x= voto di laurea; z= 1 se maschio e 0 se femmina

Effetto genere = gamma

The Nature of Qualitative Information Sometimes we can not obtain a set of numerical

values for all the variables we want to use in a model.

This is because some variables can not be quantified easily.

Examples: (a)  Gender may play a role in determining salary levels (b)  Different ethnic groups may follow different consumption patterns (c)  Educational levels can affect earnings from employment

It is easier to have dummies for cross-sectional variables, but sometimes we do have for time series as well.

Examples (a)  Changes in a political regime may affect production (b)  A war can have an impact on economic activities (c)  Certain days in a week or certain months in a year can have different

effects in the fluctuation of stock prices (d)  Seasonal effects are often observed in demand of various products

The Nature of Qualitative Information

The Use of Dummy Variables Consider the following cross-sectional model:

Yi=β1+β2X2i+ ui The constant term in this equation measures the mean

value of Yi when X2i is equal to zero. This model assumes that the constant will be the same

for all the observations in our data set. But what if we have two different subgroups (male,

female for example)?

The Use of Dummy Variables The question here is how to quantify the information

that comes from the difference in the two groups. One solution is to create a dummy variable as follows:

1 for male D =

0 for female Note that the choice of which of the two different

outcomes is to be assigned the value of 1 does not alter the results.

The Use of Dummy Variables Entering this dummy in the equation we have the

following model: Yi=β1+β2X2i+β3Di+ui

Now we have two cases Di=0 Yi=β1+β2X2i+ β3(0) +ui

Yi=β1+β2X2i+ui Di=1 Yi=β1+β2X2i+β3(1)+ui

Yi=(β1+β3) +β2X2i+ui

Intercept Dummy Variable Y

X2

Red Points: Male

Grey Points: Female

Intercept Dummy Variable Y

X2

Dummy variablesPublic school teachers’ salary by geographical region

Dummy variablesPublic school teachers’ salary by geographical region

STATA

Dummy variablesCautionIf a qualitative variable has m categories, introduce only (m-1) dummy variables to avoid DYMMY TRAP. Why?

The category for which no dummy variable is assigned is known as base, benchmark, control, comparison, reference, or omitted category

The coefficients attached to the dummy variables are known as the differential intercept coefficients because they tell by how much the value of the intercept receives from that dummy

If one needs to plug in the model all the variables as dummies, then it is mandatory to drop the constant term. The suggestion is to keep the constant term having a benchmark.

STATA

Dummy variablesRegression with a mixture of quantitative & qualitative regressors

where variables are the same as the example before and x is spending on public school for pupil in $

Dummy variables

Dummy variables as alternative to Chow test

Chow TestModello: Savings ; Income

Periodo: 1970-1995

Potenziale break strutturale nel 1982

Periodo1970-1981 Periodo1982-1995 Periodo1970-1995

n1=12 n2=14 n0(n1+n2)=26

Last equation assumes that there is no difference between the two time

Reminder

F-Test su RSS

F=(RSSr - RSSur)/k/((RSSur)/(n1+n2-2k))

this is because RSS1 and RSS2 are deemed independent, so that RSSur=RSS1+RSS2

F(k, n1+n2-2k) degrees of freedom

The Dummy Variable Test for Structural Stability

The Dummy Variable Test is much better because: •  A single equation is used to provide a set of the

estimated coefficients for tow or more structures •  Only one degree of freedom is lost for every

dummy used. •  A larger sample is being used for the estimation •  It provides us with information regarding the exact

nature of the parameter instability.

Dummy variables effect on the slope

Modello spesa per abbigliamento (y); x= reddito; z= 1 se femmina e 0 se maschio

Effetto su pendenza e intercetta. STATA for numerical expl

Dummy variables interaction effect

Where:Y = hourly wage in dollarsX = education (years of schooling)D2 = 1 if female, 0 otherwiseD1 = 1 if nonwhite and non-Hispanic, 0 otherwise

Gender and race qualitative and education quantitative.

In this model the differential effect of the gender dummy is constant across the two categories of race as well as the two categories of race (differential effect) constant across the two gender modalities: If the mean salary of male is higher, this is so whether they are nonwhite/non-hispanic or not. The same for gender.

Dummy variables interaction effect

However, in many applications it may happen that a female nonwhite/non-hispanic may earn lower wages than a male nonwhite/non-hispanic. In other words, there may be interaction between the two qualitative variables D1 and D2. Therefore their effect on mean Y may not be simply additive but also multiplicative:

Then

which is the mean hourly wage function for female nonwhite/non-hispanic workers.

Dummy variables interaction effect

Observe that

Be aware that the base category is male white or hispanic, so if all the alphas were negative, it would mean that female nonwhite/non-hispanic workers earn lower hourly wages than female or nonwhite/non-hispanic workers as compared with the base category. STATA for an expl.

Dummy variables piecewise linear regression (spline functions)

Sales commission increases linearly but in a different way (different slope) after a threshold level. This is a piecewise linear regression.

Dummy variables piecewise linear regression (spline functions)

Assuming the expected value of residuals = 0

Yi = sales commissionXi = volume of sales generated by the sales personX* = threshold value of sales also known as a knot (known in advance)D = 1 if Xi > X* = 0 if Xi < X*

Which gives the mean sales commission up to the target level X*:

beta1 gives the slope of the regression line in segment I, and beta1+beta2 in segment II

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