Time Series Econometrics - Distributed Lag Modeling Main Reading: Gujarati, Chapter 17,
Griffith, Judge and Hall (2001)
Time Series Econometrics - Distributed Lag Modeling Main Reading: Gujarati, Chapter 17,
Griffith, Judge and Hall (2001)
Discuss the assignment I.e. the data thing. Discuss the lecture notes. Discuss this weeks assignment. Begin with the missing estimations from the Simealtaneous Equation lecture.
Demonstrate that the error term and the Y are correlated in the Consumption model
Demonstrate that this leads to inconsistent standard errors
One problem with the reading for today is that Gujarati changed his chapter significantly. The core material that…
I am also adding one topic to the course I.e. a section on Panel Data Analysis.
Time and Econometrics
Time Series elements of cross-sectional (Retrospective designs)
Univariate Time Series Models
Multivariate Static Models
Multivariate Dynamic Models
Stationary Variables
Non-Stationary Variables
Panel Econometrics
Note: Most of the methods we examine are single-equation methods so bear in mind potential extensions in to multi-equation methods
Some Time Series/Stochastic Processes
Fertility in America
Vote Share of the Democrats in the 20th Century
Ice-cream Consumption
Barium Chloride Imports in to the US
Capital Expenditures and Appropriations
Fertiliy: Number of children borne to women of child-bearing age.
Introduction
Economists are often interested in variables that change across time rather than across individuals.
Simple Static models relate a time series variable to other time series variables.
The effect is assumed to operate within the period.
Think of the difference between cross sectional survey data and time series data. In the example that you analysed in the smoking assignment you are working with cross-sectional data. You could easily envisage examining smoking rates in a country over time e.g using gdp and lagged health promotion expenditures to create a dynamic policy model. The techniques used to analyse such models constitute an almost entirely different branch of statistics.
Some of the most important debates in macroeconomics hinge around the dynamic effects of policy decisions e.g. what is the effect of money supply increases on output and inflation. Specifying the lag lenghts in these relationships is an ongoing task of macroeconomics with obvious policy relevance.
Permanent versus Temporary Effects e.g. what is the effect of a lotto win on Labour Supply?
Dynamic Models
Dynamic effects.
Policy takes time to have an effect.
The size and nature of the effect can vary over time.
Permanent vs. Temporary effects.
Macroeconomics
e.g. the effect of M on Y in short run vs. the long run
this is know as impulse response function
money supply increases by 1 in year 1
returns to normal afterwards
what happens to y over time
One standard macro relation that conforms with theory is the effect of Money Supply on output. Theoretically the initial stimulus to output should be positive but not lasting.
: Consider a shock to a system. A graph of the response of the system over time after the shock is an impulse response function graph. One use is in models of monetary systems. One graphs for example the percentage deviations in output or consumption over time after a one-time one percent increase in the money stock.
Distributed Lag
yt = + 0 xt + 1 xt-1 + 2 xt-2 + et Effect is distributed through time
consumption function: effect of income through time
effect of income taxes on GDP happens with a lag
effect of monetary policy on output through time
The above model is known as a FINITE LAG MODEL. This is in distinction to an infinite lag model where the length of the lag is unknown.
Sequential estimation of the lags
The Distributed Lag Effect
Economic action
at time t Effect
at time t Effect
at time t+1 Effect
at time t+2
e.g. the ECB reduces interest rates or eases credit restrictions
Someone wins the lotto
The Distributed Lag Effect
Effect at time t Economic action
at time t Economic action
at time t-1 Economic action
at time t-2
An alternative way of thinking about it is that an observed outcome at time t is the product of a series of events through time. We can think later on about how we would know whether the actions at time t, time t-1 and time t-2 actually caused these events.
Two Questions
1. How far back?
- What is the length of the lag?
- finite or infinite
2. Should the coefficients be restricted?
- e.g. smooth adjustment
- let the data decide
Unrestricted Finite DL
Finite: change in variable has an effect on another only for a fixed period
e.g. Monetary policy affects GDP for 18 months
the interval is assumed known with certainty
Unrestricted (unstructured)
the effect in period t+1 is not related to the effect in period t
yt = + 0 xt + 1 xt-1 + 2 xt-2 + . . . +n xt-n + et n unstructured lags no systematic structure imposed on the ’s the ’s are unrestricted OLS will work
i.e. will produce consistent and unbiased estimates
Problems
1. n observations are lost with n-lag setup.
data from 1960, 5 lags in model implies earliest point in regression is 1965
use up degrees of freedom (n-k)
2. high degree of multicollinearity among xt-j’s
xt is very similar to xt-1 --- little independent information
imprecise estimates
large stn errors, low t-tests
hypothesis tests uncertain.
3. Several LHS variables
many degrees of freedom used for large n.
4. Could get greater precision using structure
Examples
See example in See example in Hill, Griffiths and Judge (Table 15.3 and 15.4).
low t statistics
strange pattern of coefficients
impulse response graph
x goes up by one unit in year 1
what happens through time?
Fertility and Personal Exemption Example
Arithmetic Lag
Still finite : the effect of X eventually goes to zero
The coefficients are not independent of each other
The effect of each lag will be less than previous one
E.G. Monetary policy in 1995 will have less of an effect on GDP in 1998 than will monetary policy in 1996
Note how this is different to the capital exp example
Arithmetic Lag Structure (impulse response function)
i i 0 = (n+1) 1 = n 2 = (n-1) n = . . . 0 1 2 . . . . . n n+1 . . . . linear
lag
structure
The Arithmetic Lag Structure
Imposing the relationship: ii = (n - i+ 1) 0 = (n+1)
1 = n
2 = (n-1)
3 = (n-2)
n-2 = 3
n-1 = 2
n = only need to estimate one coefficient, ,
instead of n+1 coefficients, 0 , ... , n . yt = + 0 xt + 1 xt-1 + 2 xt-2 + . . . +n xt-n + et
Suppose that X is (log of) money supply and Y is (log of) GDP, n=12 and g is estimated to be 0.1
the effect of a change in x on GDP in the current period is b0=(n+1)g=1.3
the impact of monetary policy one period later has declined to b1=ng=1.2
n periods later, the impact is bn= g=0.1
n+1 periods later the impact is zero
Estimation
Estimate using OLS
only need to estimate one parameter: g
Have to do some algebra to rewrite the model in form that can be estimated.
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