Econometrics Week - Time Series

Question 1

first difference

Answer 1

change in value of Y between period t-1 and period t is Y_t-Y_t-1

Question 2

lagged value

Answer 2

the value of Y in the previous period relative to hte current period, t: Y_t-1

Question 3

why do we often use logs in economic time series?

Answer 3

• many economic series exhibit growth that is approx. exponential; that is, over the long run, the series tends to grow by a certain percentage per yer on average. This implies that the log of the series grows approx. linearly
• another reason is that the sd of many economic time series is approx. proportional to its level; that is, the sd is well expressed as a percentage of the level of the series- This implies that the sd of the log of the series is approx. constant
• In either case, it is useful to transform the series so that changes in the transformed series are proportional changes in the origina series, and this is acihieved by log

Question 4

autocorrelation and autocovariance

Answer 4

Question 5

stationarity

Answer 5

• background: time series forecasts use data on the past to forecast the future. doiig so presumes that the future is similar to the past in the sense that the correlations, and more generally the distributions of the data in the future will be like they were in the past. If this wasn’t true, then historical relationships would likely not be reliable forecasts of the future
• definition: probability distribution of the time series variable does not change over time. Under the assumption of stationarity, regression models estimated using past data can be used to forecast future values
• In other words: stationarity holds when the joint distribution of (Y_s+1,…,Y_s+T) does not depend on s
• In other words, (Y₁, Y₂,…,Y_T) are identically distributed, however, they are not necessarily independent!
• reasons for non-stationarity:
  
  • unconditional mean might have a trend, e..g US GDP has a persistent upwrad trend, reflecting long-term economic growth
  • population regression coefficients change at a given point in time

Question 6

mean squared forecast error

Answer 6

• because forecast errors are inevitable (future is unknown), aim is not to eliminate errors but to make them as small as possible–>MSFE = E[Y_T+1 - Yhat_T+1|T)²] (the T+1|T means that the forecast is of the value of Y at time T+1 made using data up until T)
  
  • the MSFE is the expected value of the square of the forecast error
• the root mean squared forceast error (RMSFE) is the square root of the MSFE. Same units as Y
• if the forecase is unbiased, forecast errors have a mean of 0 and the RMSFE is the sd of the out-of-sample forecast
• large forecast erors often more costly than small ones; series of small forecast errors often only causes minor prolem, but big one can call entire forecast into question
• MSFE incorporates two sources of randomness
  
  • randomness of future value, Y_T+1
  • randomness arising from estimating forecast model
  • by adding and subtracting mu_y, if Y_T+1 is uncorrelated with muhat_y, the MSFE can be written as MSFE = E[Y_T+1-mu_y)²] + E[muhat_y-mu_y)²]; the first expression is the error the forecaster would make if the population mean were known; this captures the random future fluctuations in Y_T+1 around the population mean; the second term is the additional error made because the population mean is unknown, so forecaster must estimate it

Question 7

autoregression

Answer 7

expresses the conditional mean of a time series variable Y_t as a linear funciton of its own lagged values. A first-order autoregression uses only one lag of Y in this conditional expectation: E(Y_t|Y_t-1, Y_t-2…)=beta₀ + beta₁Y_t-1; the populatin coefficients can be setimated by OLS

Question 8

autoregressive distributed lag model

Answer 8

• autoregressive because lagged values of the dependent variable are inclded as regressors, as in an autoregression, and distributed lag because the regression also includes multiple lags of an additional predictor. In genera, an ADL model with p lags of the dependent variable Y_t and q lags of an additional predictor X_t is called an ADL(p,q) model
• Y_t = beta₀ + beta₁Y_t-1 + beta₂Y_t-2 + … + beta_pY_t-p + delta₁X_t-1 + delta₂X_t-2 + … + delta_qX_t-q + u_t
• the assumption hta the errors in the ADL model have a conditional mea of 0 given all past values of Y and X, that is, that E(u_t|Y_t-1, Y_t-2,…,X_t-1, X_t-2,…)=0 implies taht no additional lags of eiher Y or X belong in the ADL mode. In other words, the lag lenghts p and q are the true lag lengths adn the coefficients on additional lags are 0

Question 9

least squares assumptions for forecasting with time series data

Answer 9

• Y_t = beta₀ + beta₁Y_t-1 + beta₂Y_t-2 + … + beta_pY_t-p + delta₁X_t-1 + delta₂X_t-2 + … + delta_qX_t-q + u_t
• E(u_t|Y_t-1, Y_t-2, …, X_t-1, X_t-2, …)=0
  
  • u has a conditional mean of 0 given the history of all the regressors
• the random variables (Y_t, X_t) have a stationary distribution and (Y_t, X_t) and (Y_t-j, X_t-j) become independent as j gets large
  
  • stationary: so that the distibution of the time series today is the same as its distribution in the past. This assumption is a time series version of the identically distributed part of the i.i.d asssumption: the cross-sectional requirement of each draw being identically distrubiuted is replaced by the time series requirement that the joint distribution of the variables including lags, not change over time. If the time series variables are nonstationary, then one or more problems can arise in time series regression, including biased forecasts
    
    • assumption of stationarity implies taht the conditional mean for the data used to estimate the model is also the conditional mean for the out-of-sample observation of interest. Thus, the assumption of stationarity is also an assumption about external validity and it plays a role in the first least squares assumption for prediction
  • 2nd assumption is sometimes referred to as weak dependence, and it ensures that in alrge samples there is sufficient randomness in the data for the law of large numbers and the central limit theorem to hold
    
    • this replaces the cross-sectional reuqirement htat the variables be independently distributed from one observation to the next with the time series requirement that htey be independently distributed when they are separated by long periods of time
• large outliers are unlikely, i.e. X_t,…,X_kt and Y_t have nonzero, finite fourth moments
• there is no perfect multicollinearity

Question 10

estimating the MSFE method 1: by SE of the regression

Answer 10

• focuses only on future uncertainty and ignores uncertainty associated with estimation of the regression coefficients
• attachment
• because the variance of the OLS estimator is proportional to 1/T, the 2nd term in the equation is proportional to 1/T. Consequently, if the number of observations T is large relative to the number of autoregressive lags p, then the contribution of the 2nd term is small relative to the first term. That is, if T is large relative to p, simplifies to the approximation MSFE=sigma_u² This simplification suggests estimating the MSFE by MSFE_SER = s_uhat², where s_uhat² = SSR/(T-p-1), where SSR is the sum of squared residuals of the autoregression. The statistic S_uhat² is the square of the SE of the regression (SER)

Question 11

estimating the MSFE method 2: by the final prediction error

Answer 11

Question 12

estimating the MSFE method 3: by pseudo out-of-sample forecasting

Answer 12

• uses data to stimulate out-of-sample forecasting. Divide data set into two parts: Initial estimation sample is used to estimate the forecasting model, which is then used to forecast 1^st observation in the reserved sample. Next, estimation sample is augmented by 1^st observation in reversed sample, and the model is reestimated and is used to forecast the 2^nd observation in the reversed sample. This procedure is repeated until the forecast is made of the final observation in the reserved sample and produces P forecasts and thus P forecast errors. Those P forecast errors can then be used to estimate the MSFE. This method of estimating a model on a subsample of the data and then using that model to forecast on a reserved sample is called pseudo out-of-sample forecasting: out of sample because the observations being forecasted were not used for model estimation but pseudo because the reserved data are not truly out-of-sample observations.
• Compared to squared SER estimate from method 1, and final prediction error estimate in method 2, the pseudo out-of-sample estimate in this equation has both advantages and disadvantages.
  
  • advantages: does not rely on the assumption of stationarity, so that the conditional mean might differ between the estimation and the reserved samples. E.g. coefficients of the autoregression need not be the same in the two samples, and the pseudo out-of-sample forecast error need not have mean 0. Thus, any bias in the forecast arising because of a change in coefficients will be captured by MSFE_POOS but not by the other two estimators.
  • Disadvantages: more difficult to compute: estimate of MSFE will have greater sampling variability than the other two estimates if Y is, in fact, stationary (because the estimate of MSFE_POOS uses only P forecast errors); requires choosing P. The choice of P entails a trade-off between the precision of the coefficient estimates and the # of observations available for estimating the MSFE. In practice, choosing P to be 10% or 20% of the total number of observations can provide a reasonable balance between these two considerations.

Question 13

forecast uncertainty and forecast intervals

Answer 13

• In any estimation, it is good practice to report a measure of uncertainty. One measure of uncertainty of a forecast is its RMSFE. Under the additional assumption that the errors u_t are normally distributed, the estimates of RMSFE can be used to construct a forecast interval that contains the future value of the variable with a certain probability
• One important difference between forecast interval and CI: usual formula for 95% CI (estimator±1.96 SE) is justified by the central limit theorem and therefore holds for a wide range of distributions of error term. In contrast, because forecast error include the future value of the error, computing a forecast interval requires either estimating the distribution of error term or making some assumption about that distribution.

Question 14

estimating lag length using information criteria - intro

Answer 14

• how many lags to include in a time series regression? Choosin order p of an autoregression requries balancing MB of including more lags against MC of additional estimation error.
• If the order of an estimated autoregression is too low, you will omit potentially valuable information. If too high, you will be estimating more coefficients than necesary, which in turn introduces additional estimation error into your forecasts

Question 15

determining the order of an autoregression - F-statistic approach

Answer 15

start with a model with many lags and perform hypothesis tests on the final lag. If not significant, drop it and estimate the next lag. The drawback to this method is that it will tend to produce large models

Question 16

determining the order of an autoregression - BIC

Answer 16

Question 17

determining the order of an autoregression - AIC

Answer 17

Question 18

lag length selection in time series regression with multiple predictors

Answer 18

• tradeoff involved lag length choice here similar to that in an autoregression: using too few lags can decrease forecast accuracy because valuable information is lost, but adding lags increases estimation error.
• F-statistic approach: one way to determine the # of lags is to use F-statistic to test join hypotheses that sets of coefficients are 0. In general, th F-statistic method can roduce models that are large and thus have considerable estimation error.
• Information criteria: If the regression model has K coefficients (incl. intercept), BIC(K) = ln(SSR(K)/T) + K(ln(T)/T). The AIC is defined the same way, but with 2 instead of ln(T). The model with the lowest value of BIC is preferred.
  
  • two important considerations when using information criterion to estimate lag lengths
    
    • as in case for autoregression, all candidate models must be estimated over same sample - the number of observations used to estimated the model, T, must be the same for all models
    • when ther are multiple predictors, this approach is computationally demading because it requires computing many different models (many combinations of lag parameters). In practice, convenient shortcut is to require all regressors to have the same # of lags, that is to require that p=q₁=…=q_K so taht only p_max + 1 models need to be compared

Question 19

Topics of this course

Answer 19

• Introduction: time series, lags and rst dierences, logarithms and growth rates
• Autocorrelation
• Stationarity
• Dynamic models for forecasting: autoregressive model, autoregressive distributed lag model (ADL), vector autoregressive model (VAR)
• Model selection in dynamic models
• Forecasting
• Dynamic models for analyzing structural relationships
• Interpretation of regression coecients in dynamic models

Question 20

Total number of observations equals T: usually T << N, but depends on the frequency

Answer 20

why? look up

Question 21

what can time series regression models be used for?

Answer 21

• estimating dynamic causal effects
• forecasting

Question 22

OLS assumptions in time series

Answer 22

• OLS assumptions
  
  • E(u_i|X_1i,X_2i,…,X_ki) = 0
  • (Y_i,X_1i,X_2i,,…,X_ki) i.i.d.
  • Large outliers are unlikely
  • No exact linear relation between X_1i, X_2i,…, X_ki or no perfect multicollinearity
• Questions:
  
  • Are these assumptions realistic in a time series reflecting e.g. a business cycle? Likely to have serially correlated errors since u_t depends on u_t-1. This causes heteroskedasticity, and violates the i.i.d. assumption. Inconsistent but unbiased.
  • What is the consequence of heteroskedasticity?
  • What is the consequence of serial correlation?
  • What is the relevance of the fourth assumption?

Question 23

logarithms and growth rates

Answer 23

Question 24

sample autocorrelation

Answer 24

Question 25

AR(1) variance

Answer 25

Question 26

AR(1) - two different specifications

Question 27

when might stationarity not hold?

Answer 27

• one reason stationarity might not hold is that the unconditional mean might have a trend, e.g. US GDP has a persistent upward trend, reflecting long-term growth
• another type of nonstationarity arises when the population regression coefficients change at a given point in time

Question 28

What is the relevance of a(n) (asymptotically) normal distributed OLS-estimator?

Answer 28

needed for testing; lecture question, maybe see if more comprehensive answer

Question 29

two important dynamic models for estimating structural / causal relationships

Answer 29

• Partial adjustment model (slide 37)
• Error correction model (Exercise 1 Canvas)

Question 30

difference static vs dynamic model?

Answer 30

• Consider the simple linear regression model with time series data
• Y_t = beta₀ + beta₁Z_t + u_t; t = 1,…,T
• Static model: Z_t is a genuine explanatory variable (not a lag)
• Dynamic model: Z_t is a lagged regressor
  
  • Lagged value of explanatory variable X (e.g. X_t-1)
  • Lagged value of dependent variable Y (e.g. Y_t-1)
• Autoregressive economic dynamics often caused by
  
  • Delayed response
  • Adjustment costs
  • example: Y_t = beta₀ + beta₁Z_t + u_t; consider e.g. measuring wage (X) elasticity of labor demand (Y)
  • Z_t = X_t-1 because employers respond slowly to wage changes
  • Z_t = Y_t-1 because rms face adjustment costs in hiring/ring employees
• Resulting empirical model may be Y_t = beta₀ + beta₁Y_t-1 + beta₂X_t-1 + u_t

Question 31

AR, ADL, VAR models

Answer 31

pth order autoregressive model (AR(p))

• Explanation of current value Y_t by p lagged values
• Popular way of describing dynamic economic behavior; used, e.g. for modeling the evolution over time of company profits or firm size
• Useful starting point for developing models for forecasting e.g. inflation or interest rates

Autoregressive distributed lag (ADL) models

• Uses past information of additional regressor (predictor) X_t to forecast dependent variable
• Possibility to test predictive content of additional regressor X; testing H0 : delta₁ = 0,…,delta_q = 0 is labeled Granger causality test
• Note: X_t is ruled out in (14.19/15.18) but sometimes included (see e.g. slide 36)

Vector autoregressive (VAR) models

• Jointly modeling of Y_t and X_t based on only past information of both variables
• Separate equations are ADL(p,p) models
• Note: restriction on p (2x)

Question 32

OLS asumptions ADL model

Answer 32

• Assumption #1 implies that error term ut is not autocorrelated, i.e. cov(u_t,u_t-j) = 0
• Assumption #2 replaces the i.i.d. assumption made so far. Weak dependency implies that (Y_t, X_t) and (Y_t-j, X_t-j) become independent as j grows large; in other words, remote observations become independent
• Regarding VAR models these assumptions should apply equation by equation
• Extension to models with multiple X’s straightforward

Question 33

Weak independence in AR(1)

Answer 33

Question 34

How to select p and q, the number of lags of the dependent and explanatory variables in ADL models?

Answer 34

Two different approaches:

• General to specific
  
  • Start with a large enough choice for p and q in order to be sure that the error term is not autocorrelated
  • Check if some of the lags have a low t-statistic, and delete those step by step
  • Stop when all regressors are significant; test with an F test the joint signicance of the omitted dynamic terms
• Model selection criteria:
  
  • Akaike Information Criterium (AIC)
  • Bayes Information Criterium (BIC)

Question 35

AIC or BIC?

• What can you say about the relative magnitude of the AIC and BIC?
• Which one is more conservative?
• Are there any theoretical arguments in favour of AIC or BIC?

Answer 35

• Selecting the number of lags of dependent and explanatory variables can be done through an F-test or through the information criteria of Akaike and Bayes, where the lowest lag outcome is preferred. Since ln(T)>2, BIC is preferred over AIC

Question 36

Forecast error

Answer 36

Question 37

alternative way to estimate forecast error (MSFE)

Answer 37

Question 38

partial adjustment model

Answer 38

Question 39

interpretation of regression coefficients

Answer 39

• Yt = beta₀ + beta₁Y_t-1 + delta₀X_t + delta₁X_t-1 + u_t
• Static model (beta₁ = delta₁ = 0): immediate and full adjustment to the newequilibrium value within one period
• Dynamic model: adjustment from one equilibrium state to another over more than one period
• Dynamic models imply a whole pattern of effects
• Start and end points are short- and long-run effects

Question 40

short- and long-run effects

Answer 40

Question 41

Topics - (non-)stationary time series

Answer 41

• Types of nonstationarity
  
  • Trend
  • Structural break
  • Volatility (i.e. var (Yt ) constant?)
• Trends
• deterministic
• stochastic
• Random walk model
  
  • nonstationarity
  • unit root
• Detecting stochastic trends
  
  • autocorrelations
  • unit root test
• Consequences for OLS
  
  • non-normality
  • spurious regression
• Cointegration
  
  • common trends
  • testing
  • estimation
  • (V)ECM

Question 42

Trends

Answer 42

• Trend is persistent long-term movement of a variable over time
• Actual time series data often uctuate around a trend
• A priori not clear how to model trends
• two types of trends
  
  • deterministic: nonrandom function of time, e.g. a detmerinistic trend might be linear in time
  • stochastic: random and varies over time, e..g might exhibit a prolonged period of increase followed by a prolonged period of decrease; stochastic often more appropriate

Question 43

Stochastic vs deterministic trends - forecasting

Answer 43

Question 44

random walk

Answer 44

• a time series Y_t is said to follow a random walk if change in Y is i.i.d., hat is if Y_t=Y_t-1+u_t where u_t is i.i.d.
• basic idea: value of series tomorrow is value of today plus unpredictable change
• conditional mean of Y_t based on data through time t-1 is Y_t-1; that is, because E(u_t|Y_t-1,Y_t-2,…)=0, E(Y_t|Y_t-1,Y_t-2,…)=Y_t-1 in other words, if Y_t follows a random walk, then the best forecast of tomorrow’s value is its value today
• if Y follows a random walk, its variance increases over time; because it does not have a constant variance, a random walk is nonstationary
• random walk with a drift: sometimes have an obvious upward tendency, in which case the forecast must include an adjustment for the tendency of the series to increase
  
  • E(u_t|Y_t-1,Y_t-2,…)=0
  • best forecast of tomorrow is Y today + drift

Question 45

random walk vs deterministic trend

• What is the dierence between a stochastic and a deterministic trend?
• Calculate Var (Y_t) in both models.

Question 46

random walk model - unit root

Answer 46

Question 47

detecting stochastic trends

Answer 47

• Inspect sample autocorrelations: typically they tend to be very close to 1 in case of a stochastic trend
• Unit root test

Question 48

Lecture 2 question 2

Question 49

unit root test

Answer 49

Question 50

trend-stationarity

Answer 50

Question 51

Lecture 2 question 3: testing for stationarity

Answer 51

nachschauen

Question 52

consequences for OLS estimation - OLS estimator biased

Answer 52

• In case of stationary time series we have seen that OLS estimators are consistent and in large samples approximately normally distributed
• However, when we have stochastic trends OLS based inference breaks down:
  
  • OLS estimator biased
    
    • Because of the nonstationarity the basic assumption of OLS do not hold (Key concept 14.6/15.6)
    • In general there is a bias and the asymptotic distribution is no longer the normal distribution; estimate of autoregressive coefficint is biased towrads 0 (downward biased); this non-normal distribution means that conventional CIs are not valid and hypothesis tests cannot be conducted as usual
    • the downward bias of OLSestimator poses problem for forecasts: if the coefficient in an AR(1) model of the conditional mean is 1 (a unit root), then the OLS estimator willtend to take on a value less than 1, and its samplingdistribution has a mean that is less than 1
• t distributions are no longer approximately N(0; 1)
• Risk of spurious regressions

Question 53

consequences for OLS estimation - t distributions are no longer approximately N(0; 1)

Answer 53

• In case of stationary time series we have seen that OLS estimators are consistent and in large samples approximately normally distributed
• However, when we have stochastic trends OLS based inference breaks down:
  
  • OLS estimator biased
  • t distributions are no longer approximately N(0; 1)
  • Risk of spurious regressions

Question 54

consequences for OLS estimation - risk of spurious regressions

Answer 54

• In case of stationary time series we have seen that OLS estimators are consistent and in large samples approximately normally distributed
• However, when we have stochastic trends OLS based inference breaks down:
  
  • OLS estimator biased
  • t distributions are no longer approximately N(0; 1)
  • Risk of spurious regressions

Question 55

Lecture 2 question 4: regression output

Answer 55

no setting is given so more information and economic theory is needed; data shows deterministic trend

Question 56

cointegration

Answer 56

Examples:

• Permanent income hypothesis implies cointegration between consumption and income
• Money demand models imply cointegration between money, income, prices and interest rates
• Purchasing power parity implies cointegration between nominal exchange rate and foreign and domestic prices
• Covered interest rate parity implies cointegration between forward and spot exchange rates
• Growth theory implies cointegration between GDP per capita, investment rate and population
• Fisher equation implies cointegration between nominal interest rates and in
• Expectations hypothesis of the term structure implies cointegration between nominal interest rates at different maturities
• Present value model of stock prices states that a stock’s price is expected discounted present value of its expected future dividends or earnings

Question 57

testing for cointegration

Answer 57

Question 58

estimation of cointegrating relations

Answer 58

Question 59

error correction model

Answer 59

Question 60

Explain why 0 < gamma < 2 in the error correction model? What if this does not hold? That is: What is the problem? How do you deal with this problem?

Answer 60

Question 61

estimation ECM

Answer 61

Question 62

vector-error correction model

Answer 62

Question 63

properties of the forecast and error term in the AR(p) model

Answer 63

• assume that E(u_t|Y_t-1, Y_t-2,…)=0 - has two important implications
  
  • implies that best forecast of Y_T+1 based on its entire history depends on only the most recent p past value. Specifically, let Y_T|T+1=E(Y_T+1|Y_T,Y_T-1) denote the conditional mean of Y_T+1 given its entire history

Question 64

avoid problems caused by stochastic trends

Answer 64

most reliable way to handle trend in a series is to transform the series sothat it does not have the trend; if a series has a stocahstic trend, then its difference does not, e.g. if Y_t=beta₀+Y_t-1+u_t, then großdeltaY_t=beta₀+u_t is stationary

Question 65

testing for a break at a known date

Answer 65

• if the date of the break in the coefficients is known, then the null hypothesis of no break can be tested using binary variable interaction regression
• consider ADL(1,1) model: Y_t=beta₀+beta₁Y_t-1+delta₁X_t-1+gamma₀D_t(tau) + gamma₁[D_t(tau)*Y_t-1] + gamma₂[D_t(tau)*X_t-1] + u_t
• if thre is no break, then the population regression function is te same over both parts of the sample, so the terms involving the break binary variable D_t(tau) do not enter tje equation above. That is, under the null hypothesis of no break, gamma₀ = gamma₁ = gamma₂=0. Under the alternative hypothesis that there is a break, the population regression function is different before and after the break data tau, in wchih case at least one of the gamma’s is nonzero.Thus the hypothesis of a break can be tested using the F-statistic that tests the hypothesis that gamma₀ = gamma₁ = gamma₂=0 against the hypothesis that at least one of the gamma’s is nonzero.
• This approach can be modified to check for abreak iun a subset of the coefficients by including only the binary variable interactions for that subset of regressors of interest.

Question 66

testing for a break at an unknown date

Answer 66

• could for example suspoect that a break occured sometime between two dates; test for known date can be extended to handle this situation by testing for breaks at all possible dates tau between tau₀ and tau₁ and then using the largest F-statistics to test for a break at an unknown date
• since the statistic is the largest of many F-statistic, its distribution is not the same as an individual F-statistic. Instead, the ciritical values for the statistic must be obtained from a special distribution. like the F-statistic, the distribuion depends on the number of restrictions being tested, q - that is, the number of coefficients (including the intercept) that are being allowed to break, or change, under the alternative hypothesis. The distriution of the statistic also depends on tau₀/T and tau₁/T - that is, on the endpoints tau₀and tau₁ of the subsample over which the F-statistics are computed, expressed as a fraction of the total sample size
• for the large-sample approximation to the distribution of the statistic to be a good one, the subsample endpoints cannot be too close to the beginning or the end of the sample; thus statistic computed over a trimmed range, or subset, of sample
• if the statistic rejects the null hypothesis, it can mea nthat there is asingle discrete break, that there are multiple breaks, or that there is a slow evolution of the regression function

Question 67

avoiding the problems caused by breaks

Answer 67

if a distinct break occurs at a specfic date, that break will be detected with high probabiloityby the QLR statistic, and the break date can be estimated. The regression function can then be reestimated using a binary variable indicating the two subsamples associated wth this break and including interactions with the other regressors as appropriate. IF all the coefficients break, then this simplifies to reestiamting the regression using the post-break data. If there is in fact a distinct brealk, then subsequent inference on the regresio coefficients can proceed as usual - e.g. using normal critical values for hypothesis tests based on t-statistics.

Question 68

summary card chapter 15

Answer 68

copy from pdf