Urgent Econometrics test
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RegressionwithTimeSeriesData.pptxRegression with Time Series Data: Stationary Variables
When modeling relationships between variables, the nature of the data that have been collected has an important bearing on the appropriate choice of an econometric model
- Two features of time-series data to consider:
- Time-series observations on a given economic unit, observed over a number of time periods, are likely to be correlated
- Time-series data have a natural ordering according to time
There is also the possible existence of dynamic relationships between variables
- A dynamic relationship is one in which the change in a variable now has an impact on that same variable, or other variables, in one or more future time periods
- These effects do not occur instantaneously but are spread, or distributed, over future time periods
FIGURE 9.1 The distributed lag effect
Ways to model the dynamic relationship:
- Specify that a dependent variable y is a function of current and past values of an explanatory variable x
(9.1)
- Because of the existence of these lagged effects, Eq. 9.1 is called a distributed lag model
- Capturing the dynamic characteristics of time-series by specifying a model with a lagged dependent variable as one of the explanatory variables
(9.2)
Or have:
(9.3)
- Such models are called autoregressive distributed lag (ARDL) models, with ‘‘autoregressive’’ meaning a regression of yt on its own lag or lags
Ways to model the dynamic relationship (Continued):
- Model the continuing impact of change over several periods via the error term
(9.4)
- In this case et is correlated with et - 1
- We say the errors are serially correlated or autocorrelated
The primary assumption is Assumption MR4:
- For time series, this is written as:
- The dynamic models in Eqs. 9.2, 9.3 and 9.4 imply correlation between yt and yt - 1 or et and et - 1 or both, so they clearly violate assumption MR4
- A stationary variable is one that is not explosive, nor trending, and nor wandering aimlessly without returning to its mean
FIGURE 9.2 (a) Time series of a stationary variable
FIGURE 9.2 (b) time series of a nonstationary variable that is ‘‘slow-turning’’ or ‘‘wandering’’
FIGURE 9.2 (c) time series of a nonstationary variable that ‘‘trends”
Consider a linear model in which, after q time periods, changes in x no longer have an impact on y
(9.5)
-Note the notation change: βs is used to denote the coefficient of xt-s and α is introduced to denote the intercept
Model 9.5 has two uses:
Forecasting
- Policy analysis
- What is the effect of a change in x on y?
Finite Distributed Lags
Assume xt is increased by one unit and then maintained at its new level in subsequent periods
- The immediate impact will be β0
the total effect in period t + 1 will be β0 + β1, in period t + 2 it will be β0 + β1 + β2, and so on
- These quantities are called interim multipliers
- The total multiplier is the final effect on y of the sustained increase after q or more periods have elapsed
The effect of a one-unit change in xt is distributed over the current and next q periods, from which we get the term ‘‘distributed lag model’’
- It is called a finite distributed lag model of order q
- It is assumed that after a finite number of periods q, changes in x no longer have an impact on y
- The coefficient βs is called a distributed-lag weight or an s-period delay multiplier
- The coefficient β0 (s = 0) is called the impact multiplier
ASSUMPTIONS OF THE DISTRIBUTED LAG MODEL
TSMR1.
TSMR2. y and x are stationary random variables, and et is independent of
current, past and future values of x.
TSMR3. E(et) = 0
TSMR4. var(et) = σ2
TSMR5. cov(et, es) = 0 t ≠ s
TSMR6. et ~ N(0, σ2)
Consider Okun’s Law
- In this model the change in the unemployment rate from one period to the next depends on the rate of growth of output in the economy:
- We can rewrite this as:
where DU = ΔU = Ut - Ut-1, β0 = -γ, and α = γGN
- We can expand this to include lags:
- We can calculate the growth in output, G, as:
FIGURE 9.4 (a) Time series for the change in the U.S. unemployment rate: 1985Q3 to 2009Q3
FIGURE 9.4 (b) Time series for U.S. GDP growth: 1985Q2 to 2009Q3
Table 9.1 Spreadsheet of Observations for Distributed Lag Model
Table 9.2 Estimates for Okun’s Law Finite Distributed Lag Model
When is assumption TSMR5, cov(et, es) = 0 for t ≠ s likely to be violated, and how do we assess its validity?
- When a variable exhibits correlation over time, we say it is autocorrelated or serially correlated
- These terms are used interchangeably
Serial Correlation
FIGURE 9.5 Scatter diagram for Gt and Gt-1
Recall that the population correlation between two variables x and y is given by:
For the Okun’s Law problem, we have:
(9.12)
The notation ρ1 is used to denote the population correlation between observations that are one period apart in time
- This is known also as the population autocorrelation of order one.
- The second equality in Eq. 9.12 holds because var(Gt) = var(Gt-1) , a property of time series that are stationary
The first-order sample autocorrelation for G is obtained from Eq. 9.12 using the estimates:
Making the substitutions, we get:
More generally, the k-th order sample autocorrelation for a series y that gives the correlation between observations that are k periods apart is:
Because (T - k) observations are used to compute the numerator and T observations are used to compute the denominator, an alternative that leads to larger estimates in finite samples is:
Applying this to our problem, we get for the first four autocorrelations:
How do we test whether an autocorrelation is significantly different from zero?
- The null hypothesis is H0: ρk = 0
- A suitable test statistic is:
For our problem, we have:
- We reject the hypotheses H0: ρ1 = 0 and H0: ρ2 = 0
- We have insufficient evidence to reject H0: ρ3 = 0
- ρ4 is on the borderline of being significant.
- We conclude that G, the quarterly growth rate in U.S. GDP, exhibits significant serial correlation at lags one and two
The correlogram, also called the sample autocorrelation function, is the sequence of autocorrelations r1, r2, r3, …
- It shows the correlation between observations that are one period apart, two periods apart, three periods apart, and so on
FIGURE 9.6 Correlogram for G
The correlogram can also be used to check whether the multiple regression assumption cov(et, es) = 0 for t ≠ s is violated
Consider a model for a Phillips Curve:
- If we initially assume that inflationary expectations are constant over time (β1 = INFEt) set β2= -γ, and add an error term:
To determine if the errors are serially correlated, we compute the least squares residuals:
FIGURE 9.7 (a) Time series for Australian price inflation curve
FIGURE 9.7 (b) Time series for the quarterly change in the Australian unemployment rate
The k-th order autocorrelation for the residuals can be written as:
- The least squares equation is:
- The values at the first five lags are:
An advantage of this test is that it readily generalizes to a joint test of correlations at more than one lag
If et and et-1 are correlated, then one way to model the relationship between them is to write:
- We can substitute this into a simple regression equation:
We have one complication: is unknown
- Two ways to handle this are:
- Delete the first observation and use a total of T-1 observations
- Set and use all T observations
Other Tests for Serially Correlated Errors
For the Phillips Curve:
- The results are almost identical
- The null hypothesis H0: ρ = 0 is rejected at all conventional significance levels
- We conclude that the errors are serially correlated
To derive the relevant auxiliary regression for the autocorrelation LM test, we write the test equation as:
But since we know that , we get:
Rearranging, we get:
(9.26)
- If H0: ρ = 0 is true, then LM = T x R2 has an approximate χ2(1) distribution
- T and R2 are the sample size and goodness-of-fit statistic, respectively, from least squares estimation of Eq. 9.26
Considering the two alternative ways to handle :
- These values are much larger than 3.84, which is the 5% critical value from a χ2(1)-distribution
- We reject the null hypothesis of no autocorrelation
- Alternatively, we can reject H0 by examining the p-value for LM = 27.61, which is 0.000
For a four-period lag, we obtain:
- Because the 5% critical value from a χ2(4)-distribution is 9.49, these LM values lead us to conclude that the errors are serially correlated
Durbin-Watson Test: This is used less frequently today because its critical values are not available in all software packages, and one has to examine upper and lower critical bounds instead
- Also, unlike the LM and correlogram tests, its distribution no longer holds when the equation contains a lagged dependent variable
Three estimation procedures are considered:
Least squares estimation
An estimation procedure that is relevant when the errors are assumed to follow what is known as a first-order autoregressive model
A general estimation strategy for estimating models with serially correlated errors
We will encounter models with a lagged dependent variable, such as:
Estimation with Serially Correlated Errors
ASSUMPTION FOR MODELS WITH A LAGGED DEPENDENT VARIABLE
TSMR2A In the multiple regression model
Where some of the xtk may be lagged values of y, vt is uncorrelated with all xtk and their past values.
Suppose we proceed with least squares estimation without recognizing the existence of serially correlated errors. What are the consequences?
- The least squares estimator is still a linear unbiased estimator, but it is no longer best
- The formulas for the standard errors usually computed for the least squares estimator are no longer correct
- Confidence intervals and hypothesis tests that use these standard errors may be misleading
It is possible to compute correct standard errors for the least squares estimator:
- HAC (heteroskedasticity and autocorrelation consistent) standard errors, or Newey-West standard errors
- These are analogous to the heteroskedasticity consistent standard errors
Consider the model yt = β1 + β2xt + et
- The variance of b2 is:
where
When the errors are not correlated, cov(et, es) = 0, and the term in square brackets is equal to one.
- The resulting expression
is the one used to find heteroskedasticity-consistent (HC) standard errors
- When the errors are correlated, the term in square brackets is estimated to obtain HAC standard errors
If we call the quantity in square brackets g and its estimate , then the relationship between the two estimated variances is:
Let’s reconsider the Phillips Curve model:
The t and p-values for testing H0: β2 = 0 are:
Return to the Lagrange multiplier test for serially correlated errors where we used the equation:
(9.30)
- Assume the vt are uncorrelated random errors with zero mean and constant variances:
Eq. 9.30 describes a first-order autoregressive model or a first-order autoregressive process for et
- The term AR(1) model is used as an abbreviation for first-order autoregressive model
- It is called an autoregressive model because it can be viewed as a regression model
- It is called first-order because the right-hand-side variable is et lagged one period
We assume that:
The mean and variance of et are:
The covariance term is:
The correlation implied by the covariance is:
Setting k = 1:
ρ represents the correlation between two errors that are one period apart
- It is the first-order autocorrelation for e, sometimes simply called the autocorrelation coefficient
- It is the population autocorrelation at lag one for a time series that can be described by an AR(1) model
- r1 is an estimate for ρ when we assume a series is AR(1)
Each et depends on all past values of the errors vt:
For the Phillips Curve, we find for the first five lags:
For an AR(1) model, we have:
For longer lags, we have:
Our model with an AR(1) error is:
with -1 < ρ < 1
- For the vt, we have:
With the appropriate substitutions, we get:
- For the previous period, the error is:
- Multiplying by ρ:
- Substituting, we get:
(9.43)
The coefficient of xt-1 equals -ρβ2
- Although Eq. 9.43 is a linear function of the variables xt , yt-1 and xt-1, it is not a linear function of the parameters (β1, β2, ρ)
- The usual linear least squares formulas cannot be obtained by using calculus to find the values of (β1, β2, ρ) that minimize Sv
- These are nonlinear least squares estimates
Our Phillips Curve model assuming AR(1) errors is:
- Applying nonlinear least squares and presenting the estimates in terms of the original untransformed model, we have:
Nonlinear least squares estimation of Eq. 9.43 is equivalent to using an iterative generalized least squares estimator called the Cochrane-Orcutt procedure
We have the model:
(9.46)
- Suppose now that we consider the model:
(9.47)
- This new notation will be convenient when we discuss a general class of autoregressive distributed lag (ARDL) models
- Eq. 9.47 is a member of this class
Note that Eq. 9.46 is the same as Eq. 9.47 since:
- Eq. 9.46 is a restricted version of Eq. 9.47 with the restriction δ1 = -θ1δ0 imposed
Applying the least squares estimator to Eq. 9.47 using the data for the Phillips curve example yields:
The equivalent AR(1) estimates are:
- These are similar to our other estimates
The original economic model for the Phillips Curve was:
Re-estimation of the model after omitting DUt-1 yields:
In this model inflationary expectations are given by:
- A 1% rise in the unemployment rate leads to an approximate 0.5% fall in the inflation rate
We have described three ways of overcoming the effect of serially correlated errors:
- Estimate the model using least squares with HAC standard errors
- Use nonlinear least squares to estimate the model with a lagged x, a lagged y, and the restriction implied by an AR(1) error specification
- Use least squares to estimate the model with a lagged x and a lagged y, but without the restriction implied by an AR(1) error specification
An autoregressive distributed lag (ARDL) model is one that contains both lagged xt’s and lagged yt’s
Two examples:
Autoregressive Distributed Lag Models
An ARDL model can be transformed into one with only lagged x’s which go back into the infinite past:
- This model is called an infinite distributed lag model
Four possible criteria for choosing p and q:
Has serial correlation in the errors been eliminated?
Are the signs and magnitudes of the estimates consistent with our expectations from economic theory?
Are the estimates significantly different from zero, particularly those at the longest lags?
What values for p and q minimize information criteria such as the AIC and SC?
The Akaike information criterion (AIC) is:
where K = p + q + 2
The Schwarz criterion (SC), also known as the Bayes information criterion (BIC), is:
- Because Kln(T)/T > 2K/T for T ≥ 8, the SC penalizes additional lags more heavily than does the AIC
Consider the previously estimated ARDL(1,0) model:
FIGURE 9.9 Correlogram for residuals from Phillips curve ARDL(1,0) model
Table 9.3 p-values for LM Test for Autocorrelation
For an ARDL(4,0) version of the model:
Inflationary expectations are given by:
Table 9.4 AIC and SC Values for Phillips Curve ARDL Models
Recall the model for Okun’s Law:
FIGURE 9.10 Correlogram for residuals from Okun’s law ARDL(0,2) model
Table 9.5 AIC and SC Values for Okun’s Law ARDL Models
Now consider this version:
An autoregressive model of order p, denoted AR(p), is given by:
Consider a model for growth in real GDP: