Econometrics Lecture Notes: Monte Carlo simulation
In econometrics we frequently wish to explore the properties of an estimator. That is
we are not so much interested in economic theory and estimate a relationship
between Y and a vector of explanatory variables X. Instead we want to know what
happens to our estimator as sample size increases, as the variance of the error term
increases as the degree of serial correlation increases. In order to do this we make up
the data set and make up an (economic?) relationship which we then proceed to
estimate. An example will clarify, within the context of GLS and serial correlation:
Step 1: The model we specify as:
Yt = 101 + 2.3X1t -0.2X2t + ut
(1)
Step 2: Specify n (sample size to be 100)
Step3: Generate data for X1t. To do this we use a random number generator. That is
we program the computer (within, e.g. RATS) to generate 100 numbers which in turn
will be X11, X12, …….., X1,100. We need to specify the mean (μ1) and the variance
(σ21)of these numbers: X1 ~ N(μ1, σ21). We now use computers to do this, early on it is
possible a roulette wheel may have been used, hence the name. Now generate data for
X2t, t=1,..,100) using the same approach. You should probably specify a different
mean and variance. We could also make the variable trended by e.g. generating Xt2,
then calculating X2t = X2t + 0.2*t; where t goes 1,2,3,………,100.
Step 4 Generate data for u1t. This takes several stages. We assume that the error term
displays first order positive serial correlation:
ut = 0.7ut-1 + εt
(2)
where εt is a pure white noise error term. We need to specify its variance (mean will
be zero) and again we use a random number generator where εt ~N(0, σ2ε). Note N
denotes the normal distribution, i.e. our error terms will be randomly drawn from a
normal distribution with mean zero and variance σ2ε. We tend to use the normal
distribution, but you could use others. We now assume that the error term in period 1
u1 was 0 and can now generate the error terms for periods 2 to 100, by inserting the
lagged values for ut-1 and the values for εt in equation 2. Excel would do this quite
easily.
Step 5 We now know everything on the right hand side of equation (1). We can use
this to generate values for Yt, for all t.
Step 6 Estimate equation (1) using OLS, that is regress Yt on a constant, X1t and X2t.
Call this vector of estimator β0OLS, β1OLS, β2OLS for the three coefficients respectively.
(Note you will also get estimates for the variance of the error term and the first order
correlation coefficient in (2) [in this case 0.7]). Now estimate equation (2) using
GLS, where the coefficients will be termed β0GLS, β1GLS, β2GLS, respectively. We could
compare these coefficients with the known true values to see which is the closest. But
with just one comparison to make luck will play a part. Hence:
Step 7 Go back to step 3 and repeat the whole process [some would argue that we
should go to step 4, keep the X’s, change just the error term and hence Y]. Now do
this a hundred times and calculate:
100
Σ(βj0OLS – 101)/100.0
j=1
100
Σ(βj1OLS – 2.3)/100.0
j=1
100
Σ(βj2OLS – (-0.2))/100.0
j=1
where the superscript j denotes the simulation from 1 to 100. This gives you the
average bias on the 100 simulations for each of the three coefficients. For an unbiased
estimator it should be close to zero. You should do the same for the GLS estimator.
What would you expect to find? In general unbiased estimators (unless you added a
lagged dependent variable to equation (1), when together with serial correlation you
would get biased estimates when using OLS, but not GLS. However, the GLS
estimator should be more efficient, on average it should be closer to the true values.
To test this, you would calculate the variance of the 100 different estimates of the
OLS estimates for each of the three coefficients and do the same with the GLS
estimates.
8. This completes the Monte Carlo process, but you can continue. You could for
example, see what happens as you increase the sample size (at step 2), increase the
variance of the error term (at step 4), introduce heterogeneous error terms into the
model, whereby the error term is correlated with one or more of the X’s.
2SLS. If we were doing a Monte Carlo simulation with a simultaneous equation
system and comparing OLS with 2SLS, the methodology would be essentially the
same. You would need to generate X values for all the right hand side in all the
equations – remember that you need to specify a simultaneous system. You need also
to generate values for all the error terms in all the equations. However, because, e.g.,
Y1 would depend upon Y2 we could not calculate Y1 in step 5 until we know Y2. But
similarly we could not generate Y2 until we knew Y1. hence we must calculate the
Reduced Form equations, expressing Y1 and Y2 solely in terms of exogenous
variables and error terms and calculate Y1 and Y2 that way. This has been done within
the context of time series analysis because the basic example was of serial correlation.
But the techniques can be applied equally to cross section problems.
Econometrics Lecture Notes: Dummy variables & Time Trends
Shift Dummy
In the regression Yt = β0 + β1X1t + β2X2t + β3D1t, estimated 1955:Q1 to 2003Q4
D1t takes a value of 1 in 1974Q1-Q4 and 1979Q1-Q4, otherwise it takes a value of
zero. If significant it implies that in those quarters Y was β3 high(lower if negative)
than in the rest of the sample. It implies something happened in those quarters to shift
the relationship up or down. The quarters I have chosen coincide roughly with the two
oil crises of the 1979. Another example I have used in a paper, was when a civil
service strike significantly reduced the number of bankrupt firms, because the Inland
revenue and Customs and Excise are major petitioners to the Courts of bankrupt
firms. The significance of β3 is therefore a test of a structural change in the periods
specified. If D1 had taken the form, D1t =0 for all periods until 1971q1 and then a
value of one thereafter and was significant it would imply a permanent shift
upwards/downwards of the relationship and be tantamount to testing for a structural
break. In effect it changes the constant term. An alternative is called a:
Slope Dummy
In the regression Yt = β0 + β1X1t + β2X2t + β3X1tD1t, estimated 1955:Q1 to 2003Q4
If significant it implies that the coefficient on X1 was β1 for the periods when D1 was
not operative and equal to β1+ β3 in the periods when D1 was operative.
It is tremendously tempting at times when you look at a plot of the residuals and see
large positive (negative) outliers in say three successive periods or to see what looks
like a structural break to specify a dummy variable to pick this up. Some would regard
this as a form of data mining, however, if you can find a plausible event to explain
these shifts, it seems a case of when the data is informing the theory. But you must
have a plausible explanation, simply including a dummy variable because it is
significant is not acceptable.
Seasonal Dummies
When we use quarterly or monthly data we can expect there to be regular movements
in the data, based on the different times of the year. (People consume more ice cream
and lager in summer and possibly drink more coffee in winter). There are several
ways of dealing with this in econometrics. We can use seasonally adjusted data, i.e.
take data which (in general someone else) has employed some form of filter on to
change the data, taking out of it regular seasonal movements. The problem is that they
may have filtered out more than you would wish. Data is a scarce commodity, in
general filtering removes part of the signal and the econometrician’s ability to discern
the true DGP (data generating process). An alternative is to use seasonal dummies.
Thus in the following regression with quarterly data
Yt = β0 + β1X1t + β2X2t + β3SD1t + β4SD2t + β5SD3t
SD1t, SD2t, SD3t are seasonal dummies, E.g:
Quarter
…….
…….
1974Q1
1974Q2
1974Q3
1974Q4
1975Q1
1975Q2
1975Q3
1975Q4
…….
…….
SD1t
SD2t
SD3t
1
0
0
0
1
0
0
0
0
1
0
0
0
1
0
0
0
0
1
0
0
0
1
0
This indicates when estimated that Yt is β3 higher (lower) in period 1 than in period 4,
β4 higher (lower) in period 2 than in period 4 and β5 higher (lower) in period 3 than in
period 4. Period or quarter 4 is the reference quarter. None of the dummy variables
are operative in period 4. If we had included a fourth seasonal dummy variable then in
every quarter one of the dummies would have been 1, that is ΣSDjt = 1 for all t (where
the summation is across j and thus from 1 to 4). This coincides with the constant term
and the regression would be exactly collinear we would not be able to invert (X’X) to
obtain (X’X)-1X’Y. Sometimes computers do give results, but they are nonsense. This
is known as ‘the dummy variable trap’. There are other forms, e.g. if we divide people
into rural, town and city dwellers and have dummies for all three, we run into the
dummy variable trap.
There is a further method of dealing with seasonality. If we take the above equation
and lag it four periods:
Yt-4 = β0 + β1X1t-4 + β2X2t-4 + β3SD1t-4 + β4SD2t-4 + β5SD3t-4
Subtract this from the previous equation:
Yt -Yt-4 = β0-β0 + β1[X1t -X1t-4] + β2[X2t -X2t-4] +
β3[SD1t -SD1t-4] + β4 [SD2 -SD2t-4] + β5[SD3t - SD3t-4]
= β1[X1t -X1t-4] + β2[X2t -X2t-4]
That is regressing annual changes of left hand side variable on right hand side
variables [with no constant term] removes problem of seasonality.
Time Trend
In the Cobb Douglas production function:
Yt = AKtLtβeγt
We can take [natural] logs to base e:
lnYt = lnA + lnKt + βlnLt + γt
t is an example of a time trend, it increases by 1 every period, 1,2,3,…………….,n. It
reflects the impact on the left hand side variable of something which is changing in a
steady manner over time and which we cannot otherwise model. In the above case it is
the impact of productivity growth and γ is an estimate of productivity growth.