Econometrics Flashcards
LLN?
Law of large numbers - As the sample size increases, the sample mean will converge to the population mean.
CLT?
Central Limit Theorem - As the sample size increases, the sample converges to a normal distribution.
I.I.D?
Independent and identical distribution.
What does stochastic mean?
It is used to signify that a variable picked is random.
Why are time series data considered random even thought they are picked at a specific time?
Because any change in history could have altered the realised value of the process. By this logic, every realised value is therefore random.
Temporal ordering?
The ordering of the values is based on the time period.
Contemporaneous meaning?
Existing at or occurring in the same period of time
Why do we sometimes use the term ‘static model’ when referring to time series
We are modelling a contemporaneous relationship between y and z, using a certain time as the base year for example and representing changes from this base.
How should we therefore think of time randomness in time series? (1. outcomes, 2. time series, 3. sample)
The outcomes of economic variables are uncertain, they should therefore be modelled as random variables. Time series are sequences of random variables (= stochastic process). A sample is the one realised path of the time series out of the many possible paths the stochastic process could have taken.
FDL meaning and whether it is a static or dynamic model?
A Finite Distributed Lag model is a dynamic model where we allow one or more variables to affect y with a lag.
FDL and its orders?
The order of the FDL will determine how many lags an impact will have an effect on the aggregate value for. A Lag of three will mean that a value will add to the aggregate y for three periods after its original effect. (see notes for example)
What is another name for δ0 (gamma 0) in FDL models and how is it found? What is its meaning?
The impact multiplier/ propensity. Found by the difference between y and y. It is the immediate change in y due to a one unit increase in the parameter.
What is the LRP or LRM?
The Long-run Propensity or Long-Run Multiplier. It is the permanent increase in y given a permanent increase in z.
What is the likely reason for z to be omitted?
Delta<0> (the impact propensity) = 0.
Dependent variable?
The y value, the explained variable.
Independent Variable?
x. The explanatory variable.
X (bold) meaning?
The collection of all independent variables for all time periods. Useful in time series to think of it as n rocks and k columns.
What are the five classical assumptions for time series?
TS1: Linear in parameters TS2: No perfect collinearity TS3: Strict Exogeneity TS4: Homoskedasticity TS5: No Autocorrelation
TS3 intuitive understanding?
Strict exogeneity/ zero conditional mean. Implies that the error at time t is uncorrelated with each explanatory variable in every time period. So the expectation of U at time time, given all X, =0.
TS1-TS3?
Unbiasedness of the OLS estimator.
Contemporaneous Exogeneity intuitive understanding and application to consistency and unbiasedness?
The error term at time t is uncorrelated with the explanatory valuables also dated at time t. It implies that u and the explanatory variables are contemporaneously uncorrelated. Contemporaneous exogeneity is all that is needed to prove the OLS estimator is consistent but not enough for unbiasedness.
When would TS5 fail and what would be the explanation of failure?
The error would suffer from autocorrelation or serial correlation, meaning that say the error in the current period is correlated with the previous period (so if the previous period was positive, the chance of the current period being positive is higher)
BLUE?
Best in linear unbiased estimators. Requires Gauss Markov Assumption (TS1-TS5).
Gauss-Markov Theorem?
Under TS1-TS5, the OLS estimators are the best linear unbiased estimators conditional on X.
Why are dummy variables useful in time series?
A dummy variable will represent whether, in each time period, a certain event has occurred.
Detrending a variable?
Can be done by regressing that variable with a linear time trend and computing its residual. Its fundamental use is create a new variable that trends with time, thereby removing the part of the given independent variable/s trending with time.
Why might we want to detrend?
Regressing retreaded variables yield the same slope estimates on x as one would when time trend is added. Generally, if you have a variable that is trending, it is a good idea to add a time trend.
A time trending variable?
A variable that appears to increase in time with no reason aside other factor being the causal reason for this
Why might exponential time trends be more realistic?
It is often the case that growth is more exponential, whereby it is increasing at an increasing rate.
Pareto rule?
80/20 almost natural law. Roughly states that 80% of the effects come from 20% of the causes.
Spurious regression problem and how to correct for it?
Occurs when a relationship between two or more trending variables (in time) simply because each is growing over time. Adding a time trend eliminates this problem.
When might a time trend variable be added?
If a trend term is statistically significant and the results change in important ways when a time trend is added to a regression
Seasonal dummies?
Dummy variables used to alter the dependent variable at certain times of the year..
Dummy trap?
Violates ‘No perfect correlation’. Occurs when the base year variable isn’t omitted in the equation.
De-seasonalising the data?
Similar to detrending the data. It makes a new dummy variable to account for seasonal changes in y.
Week3….
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When is a time series stationary?
A stationary time series process is one whose probability distributions are stable over time in the following sense: If we take any collection of random variables in the sequence and then shift that sequence ahead h time periods, the joint probability distribution must remain unchanged.
What two assumptions does stationaity simplify?
The LLN and CLT.
What does stationaity in simple terms have to do with?
The joint distributions of a process as it moves through time. The mean, variance and autocorrelation structure do not change over time.
When is a time series weakly dependent?
A time series consisting of elements that are generally less correlated the further away they are from each other. Basically nearly independent, so that correlation between x and x goes to 0 sufficiently quickly as h approaches infinity.
Asymptotically uncorrelated?
With respect to weakly dependent time series, stating that as h tends to infinity, the joint correlation between two sets of values along the trend with the same sample size will be uncorrelated. Covariance stationary processes are said to be asymptotically uncorrelated.
When is an OLS estimator consistent?
Under assumptions TS.1-TS.3
When are OLS estimators asymptotically normal?
Under assumption TS.1-TS.5
EMH?
Efficient Market Hypothesis - States that the most recent information contains all that is required to make prediction.
Why does it make sense that time series data are weakly dependent/ correlated?
Otherwise all the data would give us no information about future data. This dependence will fall as t approaches infinity though.
Noise process?
Just a time series of i.i.d variables. This enables it to be modelled in processes.
What would be the assumptions of a MA(1) process?
There will be joint correlation between two observations next to each other in a time series, but any association between variables two or more period apart will be independent. e will be independent across t, meaning, time is independent across t with a more than 1 period difference. Therefore MA(1) is weakly dependent, stationary and the law of large numbers and the central limit theorem can be applied for x.
What is the stability condition with regards to an AR process?
The crucial assumption for the weak dependence of AR1. The stability condition would be the |Row| < 1, which intuitively means that ———–
When would we determine that a series is nonstationary?
If it violates one of the stationary principle.
What are the properties of a covariance stationary process and when is this definition used?
1.E (yt) = 𝛍
2. Var(yt) = 𝛔^2
3. Cov(yt,ys) = Cov(yt+h, Ys+h).
Used when there is a finite second moment.
A stationary stochastic process?
xt has the same distribution for all t. It is identically distributed. The stationarity definition means that the covariance between two sets will be dependent on h.
What is stationarity generally?
Involves the joint distributions of a process as it moves through time.
Weak dependency?
Assumes that the relationship between xt and xt+h are ‘almost independent’ as h increases without a bound.
We assume a series is weakly dependent if the covariance
Why is weak dependency important for regression analysis?
It replaces the assumption of random sampling in implying that the LLN and CLT hold.
In which way is a time series data expected to be different to cross sectional data?
d) Time series tend to be serially-correlated
If β₀ = 0 but β₁ ≠ 0 and/or β₂ ≠ 0, is the model static?
No
If β₀ ≠ 0 but β₁ = β₂ = 0, is the model dynamic?
No
What is the impact propensity that captures the immediate change in y due to a one-time unit increase in x?
β₀ = 0
What is the long-run propensity that captures the permanent change in y due to a permanent unit increase in x?
β₀ + β₁ + β₂
What type of exogeneity is sufficient to fulfil the Gauss Markov assumption require on xt?
Strict exogeneity
Is it true that contemporaneous exogeneity implies strict exogeneity?
No
Is heteroskedasticity of the error term required for Gauss Markov assumption with time series?
Yes
Is strict exogeneity necessary for consistent estimation of an OLS estimator?
No, only contemperaneous exogeneity (along with TS1’ and TS2’)
The statement, “An R² is harder to compute with cross sectional data compared to time series data”, is
not true
Which of the following properties is/are necessary for a time series to be a stationary process {Xt}?
a) E(Xt)=E(Xt+h) for all t,h
b) Var(Xt)=Var(Xt+h) for all t,h
c) Cov(Xt,Xt+h)=Cov(Xs,Xs+h) for all t,s,h
d) All of the above.
a) E(Xt)=E(Xt+h) for all t,h
b) Var(Xt)=Var(Xt+h) for all t,h
c) Cov(Xt,Xt+h)=Cov(Xs,Xs+h) for all t,s,h
d) All of the above.
d
Is it true that an autoregressive process of order 1 is stationary?
Cannot say with the given information
Is it true that a trend stationary time series is stationary?
No
Which processe is the most appropriate for modelling time series that have correlation persistence across a finite number of time periods?
Moving average
How is a weakly dependent process similar to a noise process?
Correlations between variables that are far apart in time have either no or weak correlation
How is a trend stationary time series similar to a noise process?
They have constant variance
Is it true that an autoregressive process of order 1 is asymptotically uncorrelated?
Cannot say with the given information.
Is it true that a moving average process of order 1 is asymptotically uncorrelated?
Yes
Let {yt} be a time series on stock returns. What does the efficient market hypothesis (EMH) imply?
E(yt|y₁,…,yt-1) = E(yt)
Not:
E(yt|y₁,…,ys) = E(ys) for 1
Which of the following regressions may be useful for studying the EMH?
yt = α + βyt-1 + ut
ΔYt = 0.513 - 0.106ΔXt + 0.076ΔXt-1
(0.218) (0.437) (0.013)
where ΔXt = Xt - Xt-1 and ΔYt = Yt - Yt-1.
Suppose you are told that the first order autocorrelations for {Yt} and {Xt} are 0.901 and 0.923 respectively. What you may expect to observe if you run a linear regression of Yt on Xt?
The slope parameter is significant and R-squared may be very high
ΔYt = 0.513 - 0.106ΔXt + 0.076ΔXt-1
(0.218) (0.437) (0.013)
where ΔXt = Xt - Xt-1 and ΔYt = Yt - Yt-1
What may a regression on first differences instead of levels achieve?
First differences may be weakly dependent
ΔYt = 0.513 - 0.106ΔXt + 0.076ΔXt-1
(0.218) (0.437) (0.013)
where ΔXt = Xt - Xt-1 and ΔYt = Yt - Yt-1.
What is the expected permanent change a unit increase in ΔXt has on ΔYt?
-0.03
What is the purpose of the Breusch-Godfrey test?
Test if the error terms are correlated to one another
