Lecture Notes 9 Econ B2000, MA Econometrics Kevin R Foster, CCNY Fall 2012

 

 

 

Instrumental Variables Regression

 

• valid instrument, some  for regression
• relevance:  and
• exogeneity:
• instrument explains X but NOT Y – can be excluded from list of variables explaining Y
• Two-Stage Least Squares (TSLS or 2SLS)
• ,
• get  and regress  on
• General Case:
• X are endogenous regressors
• W are exogenous regressors
•  are instruments
• if m>k then "overidentified"; if m=k then just identified; if m<k then unidentified
• still need:
• X, W, Z are all i.i.d. with fourth moments
• W not perfectly collinear
• Instrument Relevance and Exogeneity
• Two-Stage Least Squares:
• regress X on Z to get
• then regress Y on W and
• Evaluating Instruments in the Real World
• Weak instruments: check first-stage regression F-stat bigger than 10?
• Examples:
• cigarette tax to find effect of price
• prison capacity in place of jail terms
• random variation in births for class size
• geography for heart attack treatment
• number of immigrants 10 years ago for immigrant increase
• Mariel boatlift, other policy shifts
• deployment of police after 9/11 to estimate effects of police on crime
• Bad examples of poor instruments:
• weak instrument: month of birth on wage earnings
• Many bad examples where instruments needed:
• wage explained by schooling
• health insurance explained by wage
• wage explained by weight (discrimination against fat people?) vs wage explained by race/ethnicity (discrimination against minorities)
• Heckman 2-step for 2-part questions: first, "yes or no?"; next "how much?"  Like 2SLS but first stage is a probit!  Again need an exclusion restriction, some variable that explains the first step but not the second.
• Two-Stage Least Squares in SPSS:
• run first-stage regression, save the predicted values
• use predicted values in the second-stage prediction

 

 

 

Experiments and Quasi-Experiments

• ideal: double-blind random sort into treatment and base sets
• differences estimator
• Problems can be internal:
• incomplete randomization
• failure to follow treatment protocol
• attrition
• experiment (Hawthorne) effects
• or external
• non-representative sample
• non-rep program
• treatment/eligibility
• general equilibrium effects

 

 

 

Time Series

Basic definitions:

• first difference VYt = Yt – Yt-1
• percent change is  and is approximately equal to ln(Yt) – ln(Yt-1) – this log approximation is commonly used
• lags: the first lag of Yt is Yt-1; second lag is Yt-2, etc.
• Autocorrelation: how strong is last period data related to this period?  The autocorrelation coefficient is  for each lag length, j.  Sometimes plot a graph of the autocorrelation coefficients for various j.
• Stationarity: a model that explains Y doesn't change over time – the future is like the past, so there's some point to examining the past – a crucial assumption in forecasting!  But this is why we usually use stock returns not stock price – the price is not likely stationary even if returns are.
• If autocorrelations are not zero, then OLS is not appropriate estimator if X and Y are both time series!  The standard errors are a function of the autocorrelation terms so cannot properly evaluate the regression.
• Seasonality is basically a regression with seasons (months, days, whatever) as dummy variables.  So could have  - remember to leave one dummy variable out!  Or .

 

Types of Models

• AR(1) – autoregression with lag 1
• Forecast error is one-step-ahead error
• Note that can re-write the AR(1) equation, by substituting , as , then substitute in for , and so on.  So the current value is a function of all past error terms, .  Note that as long as , the last term drops and the sums converge  as .
• Reminder of convergent series: look at , note that .  Add and subtract  and fiddle the parentheses to write .  Notate that ugly term , then the equation says that .  Solve, , and .  Substitute this into the previous equation for Y_t
• .  As , the first term goes to , the last term goes to zero, and the middle term is .
• If  then none of the terms converge – the model becomes a random walk or integrated with order 1, I(1) or has a unit root.  (Can test for this, most common is Augmented Dickey-Fuller ADF.)
• Random walk means that AR coefficients are biased toward zero, the t-statistics (and therefore p-values) are unreliable, and we can have a "spurious regression" – two time series that seem related only because both increase over time
• AR(p) – autoregression with lag p
• ADL(p,q) – autoregressive distributed lag model with p lags of dependent variable and q lags of an additional predictor, X.
• Need usual assumptions for this model
• Lag length?  Some art; some science!  Various criteria (AIC, BIC, given in text) to select lag length.
• Granger Causality – jargon meaning that X helps predict Y; more precisely X does not Granger-cause Y if X does not help predict Y.  If X does not help predict Y then it cannot cause Y.
• Trends provide non-stationary models
• Random walk non-stationary model:
• Breaks can also give non-stationary models
• test for breaks, sup-Wald test
• Can model time series as regression of Y on X, of ln(Y) on ln(X), of DY on DX, or of %DY on %DX (where, recall, %DY = DlnY since the derivative of the log is the reciprocal) – this is where the art comes in!
• Distributed lag models can be complicated (Chapter 15) and so we want at a minimum Heteroskedasticy and Autocorrelation Consistent (HAC) errors – like the heteroskedasticity-consistent errors before (Newey-West)
• VAR – Vector AutoRegression, incorporate k regressors and p lags so estimate as many as k*p coefficients – works best with lots of data!
• GARCH models – Generalized AutoRegressive Conditional Heteroskedasticity models – allow the variance of the error to change over time, depending on past errors – allows "storms" of volatility followed by quiet (low-variance)

 

 

Factor Analysis

Another common procedure, particularly in finance, is a factor analysis.  This asks whether a variety of different variables can be well explained by common factors.  Sometimes when it's not clear about the direction of causality, or where the modeler does not want to impose an assumption of causality, this can be a way to express how much variation is common.  As an example. one price that people often see, which changes very often, is the price of gasoline.  If you have data on the prices at different gas stations over a long period of time, you would basically see that while the prices are not identical, they move together over time.  This is not surprising since the price of oil fluctuates.  There might be interesting variation that at some times certain stations might be more or less responsive to price changes – but overall the story would be that there is a common influence.

 

Factor Analysis (and the related technique of Principal Components Analysis, PCA) are not model-based and can be useful methods of exploration.  An example might be the easiest way to see how it works.

 

I have data from the US Energy Information Administration (EIA) on the spot and futures prices of gasoline from 2005-2012. (Spot prices are the price paid for delivery today; futures prices are prices agreed now for delivery in a few months.)  The prices also differ depending on where they were delivered since the price of gasoline varies over different parts of the country – although we usually only hear about it when something goes wrong with the system (e.g. a refinery must be closed or a storm damages a port or pipeline) and the variation becomes large.  We would have every reason to expect that these prices ought to be highly correlated.  With SPSS we can use "Analyze \ Dimension Reduction \ Factor".  This gives us output like this:

 

Total Variance Explained
Component	Initial Eigenvalues			Extraction Sums of Squared Loadings
	Total	% of Variance	Cumulative %	Total	% of Variance	Cumulative %
1	5.908	98.470	98.470	5.908	98.470	98.470
2	.057	.952	99.422
3	.019	.320	99.742
4	.010	.172	99.914
5	.003	.055	99.969
6	.002	.031	100.000
Extraction Method: Principal Component Analysis.

 

If you've taken linear algebra you'll recognize the eigenvalue as determining the common variation.  In this case, looking at the third column, "% of Variance," we see that the first component explains 98.470% of the variation in the 6 variables.  The additional factors (up to 6) make little additional contribution.  So in this case it is reasonable to represent these 6 price series as being mostly (more than 98%) explained by a single common factor.

 

So from the output,

 

Component Matrixa
	Component
	1
Futures1Month	.996
Futures2Months	.997
Futures3Months	.995
Futures4Months	.989
NYGasSpot	.993
GulfGasSpot	.985
Extraction Method: Principal Component Analysis.
a. 1 components extracted.

 

This gives the "loading" of the factor on each of the variables, which is the correlation of the factor with the variable.  In this case it is difficult to perceive much difference.

 

For another example, consider daily data on US interest rates at various maturities (from the Federal Reserve website). The maturities are the Fed Funds (overnight), 4 weeks, 3 and 6 months, 1 year Treasuries, and swap rates at 1, 2, 3, 4, 5, 7, 10, and 30 years.  The output shows,

Total Variance Explained
Component	Initial Eigenvalues			Extraction Sums of Squared Loadings
	Total	% of Variance	Cumulative %	Total	% of Variance	Cumulative %
1	11.035	84.882	84.882	11.035	84.882	84.882
2	1.406	10.816	95.698	1.406	10.816	95.698
3	.448	3.450	99.148
4	.058	.444	99.592
5	.031	.235	99.827
6	.011	.086	99.912
7	.006	.046	99.958
8	.004	.028	99.986
9	.001	.009	99.996
10	.000	.003	99.999
11	.000	.001	100.000
12	2.848E-05	.000	100.000
13	1.895E-05	.000	100.000
Extraction Method: Principal Component Analysis.

We see that two principal components explain over 95% of the variation.

 

The initial component correlation is

 

Component Matrixa
	Component
	1	2
Federal funds effective rate	.903	-.369
3-month Treasury bill secondary market rate   discount basis	.906	-.369
6-month Treasury bill secondary market rate   discount basis	.944	-.317
4-week Treasury bill secondary market rate   discount basis	.867	-.393
1-year Treasury bill secondary market rate^  discount basis	.966	-.242
Rate paid by fixed-rate payer on an interest rate swap with maturity of one year.	.913	-.240
Rate paid by fixed-rate payer on an interest rate swap with maturity of two year.	.972	-.041
Rate paid by fixed-rate payer on an interest rate swap with maturity of three year.	.975	.129
Rate paid by fixed-rate payer on an interest rate swap with maturity of four year.	.961	.239
Rate paid by fixed-rate payer on an interest rate swap with maturity of five year.	.945	.314
Rate paid by fixed-rate payer on an interest rate swap with maturity of seven year.	.917	.397
Rate paid by fixed-rate payer on an interest rate swap with maturity of ten year.	.886	.450
Rate paid by fixed-rate payer on an interest rate swap with maturity of thirty year.	.807	.477
Extraction Method: Principal Component Analysis.
a. 2 components extracted.

 

Which is a bit difficult to interpret.  We can ask SPSS to rotate the factors (click the button for "Rotation" and check "Varimax" which is the most common).  For those remembering some linear algebra, this is an orthogonal rotation.  The point of rotation is to help interpret the factors.  A rotated factor loading is:

 

Rotated Component Matrixa
	Component
	1	2
Federal funds effective rate	.912	.347
3-month Treasury bill secondary market rate   discount basis	.914	.350
6-month Treasury bill secondary market rate   discount basis	.906	.414
4-week Treasury bill secondary market rate   discount basis	.902	.305
1-year Treasury bill secondary market rate^  discount basis	.870	.483
Rate paid by fixed-rate payer on an interest rate swap with maturity of one year.	.831	.449
Rate paid by fixed-rate payer on an interest rate swap with maturity of two year.	.738	.634
Rate paid by fixed-rate payer on an interest rate swap with maturity of three year.	.624	.760
Rate paid by fixed-rate payer on an interest rate swap with maturity of four year.	.538	.831
Rate paid by fixed-rate payer on an interest rate swap with maturity of five year.	.475	.875
Rate paid by fixed-rate payer on an interest rate swap with maturity of seven year.	.398	.916
Rate paid by fixed-rate payer on an interest rate swap with maturity of ten year.	.340	.934
Rate paid by fixed-rate payer on an interest rate swap with maturity of thirty year.	.263	.900
Extraction Method: Principal Component Analysis.  Rotation Method: Varimax with Kaiser Normalization.
a. Rotation converged in 3 iterations.

 

Where we can clearly see that the first component is a short-term innovation with effects that die off over longer maturities while the second component is a long-term innovation with small effects on short rates but larger effects on long-term rates.  This interpretation is convenient and helps us understand how interest rates in the US move.  If one were hedging interest rate risk, there are a wide variety of instruments but two main components so a firm could hedge 95% of its exposure with two securities.

 

 

 

Econometrics goes on and on – there are thousands of techniques for new situations and new conditions, especially now that computing power quickly increases the amount of calculations that can be done.  There is so much to learn!