Time Series Econometrics

Question 1

What 3 components could a time series (trend) be broken down to and what are their definitions?

Answer 1

The Trend: The long term behaviour of the time Series.
The Cyclical: The regular periodic movements.
The Irregular: a stochastic process that econometricians hope to estimate.

Question 2

What would we create difference equations to represent?

Answer 2

The 3 components of a Time Series: The Trend, Seasonal and Irregular.

Question 3

For a time series, what do the difference equations express?

Answer 3

The value of a variable as a function of its own lagged values, time, and other variables.

Question 4

What are the trend and seasonal terms both functions of?

Answer 4

Time.

Question 5

What is the irregular term a function of?

Answer 5

Its own lagged value and of the stochastic variable

Question 6

What does the Random Walk Hypothesis require in order to be a stochastic difference equation?

Answer 6

That α(0) = α(1) = 0 and ε(t+1) has a mean (expected value) of 0.

Question 7

ARMA and ARIMA?

Answer 7

Autoregressive (integrated) Moving Average.

ARIMA has a unit root for p or q outside of the unit circle.

Question 8

When is a sequence (ε(t)) a white noise process?

Answer 8

If each blue in the series has:
1. A mean of 0.
2. A constant variance.
3. Is uncorrelated with all other realisations.
See notes for the formulation of this.

Question 9

Stationarity in time series?

Answer 9

A stationary time series is one whose statistical properties such as mean, variance, autocorrelation, etc. are all constant over time.

Question 10

How do we describe random variables?

Hint: (PDFs)

Answer 10

They are described by Probability Distribution Functions.

Question 11

What are the basic features (‘moments’) of pdfs? (5).

Answer 11

1. Location (expected value): E(y)
2. Dispersion (variance): V(y) & Standard Deviation: SD(y)
3. Skewness: S(y)
4. Kurtosis: K(y)

Question 12

Basic Features of PDFs: Expected Value Notation

Answer 12

See Econometrics Brainscape Companion 1

Question 13

Basic Features of PDFs: Variance Notation

Answer 13

See Econometrics Brainscape Companion 2

Question 14

Basic Features of PDFs: Standard Deviation Notation

Answer 14

See Econometrics Brainscape Companion 3

Question 15

Basic Features of PDFs: Skewness Notation

Answer 15

See Econometrics Brainscape Companion 4

Question 16

Basic Features of PDFs: Kurtosis Notation

Answer 16

See Econometrics Brainscape Companion 5

Question 17

Basic Concepts: Covariance Formula?

Answer 17

See Econometrics Brainscape Companion 6

Question 18

Basic Concepts: Correlation Formula?

Answer 18

See Econometrics Brainscape Companion 7

Question 19

Basic Concepts: E(a + bX)?

Answer 19

See Econometrics Brainscape Companion 8

Question 20

Basic Concepts: Var(a + bX)?

Answer 20

See Econometrics Brainscape Companion 9

Question 21

Basic Concepts: Var(aX +/- bY)?

Answer 21

See Econometrics Brainscape Companion 10

Question 22

Basic Concepts: E(XY)?

Answer 22

See Econometrics Brainscape Companion 11

Question 23

Basic Concepts: E(Y)^2?

Answer 23

See Econometrics Brainscape Companion 12

Question 24

in time series econometric models, what do ‘t’ and ‘h’ denote?

Answer 24

t = time
h = length of a time period.

Question 25

How do we get to a solution by iteration? Use an example of y0 and y1. for the difference equation yt = a0 + a(t)y(t-1)

Answer 25

We would simply calculate the function for y1, then ululate y2 (which will incorporate y(1).

Question 26

Stochastic Process?

Answer 26

Stochastic refers to a randomly determined process. It refers to a single occurrence of a random variable.

Question 27

A reduced-Form equation?

Answer 27

One that expresses the value of a variable in terms of its own lags, lags of other endogenous variables, current and past values of exogenous variables, and disturbance terms.

Question 28

Univariate Reduced Form Equations?

Answer 28

When the dependent variable is expressed solely as a function of its own lags and disturbance terms. It is useful as it allows future predictions purely based on the current and past realisations.

Question 29

The dependent Variable Axis?

Answer 29

The Y Axis. It is the effect or outcome due to the independent variable (the cause).

Question 30

Why is iteration used?

Answer 30

To enable us to attain solutions to difference equations.

Question 31

A structural équation?

Answer 31

Expresses the endogenous variable, it, as being dependent on the current realisation of another endogenous variable, ct.

Question 32

The forcing process/factor?

Answer 32

xt

The function of time, current and lagged values of there variables, and/or stochastic disturbances.

Question 33

How would you iterate with an initial condition?

Answer 33

Start at y0 and iterate forwards or backwards given the first order difference equation. Keep going until a solution can be found for the sum of all yt’s.

Question 34

Why do we have a problem in iteration when we lack an initial condition?

Answer 34

We have no known value, so cannot produce a solution, only a equation. We have to iterate in a different way. See chapter 1.3 notes.

Question 35

If we have the geometric series: r^0 + r^1 + r^2 + … + r^n, what is the value of the series as n approaches infinity? When is this useful

Answer 35

1/(1-r)

When looking at how to attain a solution from an equation when wit iteration when we have no initial condition (e.g. y0)

Question 36

What is the issue with iteration when |α| > 1?

Answer 36

The α1 value does not converge as m increases, leading to an ever growing value. As such, the sequence is not convergent and an initial condition (y0) will be required in order to provide a solution for a given first-difference stochastic equation.

Question 37

Define the Lag Operator

Answer 37

The Lag Operator (L) is defined to be a linear operator such that for any value of yt, Li y(t) is equivalent to y(t-i)

So L^I preceding y(t) simply means to lag y(t) by I periods.

Question 38

Standard deviation, variance and covariance?

Answer 38

Standard Deviation: 𝛔
Variance: 𝛔^2 (the SD squared)
Cov (X,Y) = 𝛔xy

Question 39

Lag Operators?

Answer 39

Used to provide concise notation.
y(t-1) = L
y(t-2) = L^2 etc.

Question 40

Lag Operators: How would you write the 4th-th order equation y(t) = 𝛔(0) + 𝛔(y-1) + … + 𝛔(p)y(t-p) + ε(t)? Long form and Compactly?

Answer 40

(1 - 𝛔(1)L - 𝛔(2)L^2 - … - 𝛔(p)L^p) y(t) = 𝛔(0) + ε(t)

or Compactly:
A(L) = 𝛔(0) + ε(t)
where A(L) is the polynomial: 1 - 𝛔(1)L - 𝛔(2)L^2 - … - 𝛔(p)L^p

Question 41

Explanation of A(L)?

Answer 41

A(L) is a polynomial of order (decided by the max number of lags).

Question 42

What is the Box Jenkins methodology? What are the models addressed by this methodology called?

Answer 42

A method for estimating time-series models of the form:

y(t) = 𝛔(0) + 𝛔(1)y(t-1) + … + 𝛔(p)y(t-p) + ε(t) + β(1)ε(t-1) + β(q)ε(t-q)

Question 43

What are the models addressed by this methodology called?

Answer 43

ARIMA (Auto Regressive Iterated Moving Average) time series models.
y(t) = 𝛔(0) + 𝛔(1)y(t-1) + … + 𝛔(p)y(t-p) + ε(t) + β(1)ε(t-1) + β(q)ε(t-q)

Question 44

Notation For Discrete and Continuous Time

Answer 44

Discrete: Xt
Continuous: X(t)
Actual use of brackets - unlike it usually representing the subscript version. Assume the t’s are also subscripted.

Question 45

Notation for a Discrete Random Variable?

Answer 45

R(v)

Question 46

What is the white noise process? (also use notation/ formulas)

Answer 46

In discrete stochastic time series model, a sequence {ε(t)} is a white noise process if each value in the series has the following properties:
1. A mean of 0
E(ε(t)) = E(ε(t-1)) = … = 0
2. A constant variance
E(ε(t)^2) = E(ε(t-1)^2) = … = 𝛔^2
Or Var(ε(t)^2) = Var(ε(t-1)^2) = … = 𝛔^2
3. Is uncorrelated with all other realisations
E(ε(t)ε(t-s)) = E(ε(t-j)ε(t-j-s)) = … = 0 for all j and s.

Question 47

What would we call a stationary ARIMA model?

Answer 47

An ARMA (Auto Regressive Moving Average) Model.

Question 48

What may be the confusion with discreteness in reference to ARIMA and ARMA models?

Answer 48

We assume that time is discrete, but not necessarily y(t). E.g., the rainfall in Scotland may be a continuous variable (y(t)), but the time period will be discrete.

Question 49

Once we have realised the first t observations, how would we write the expected value of Y(t+i)?

Answer 49

This would be the conditional expected value:

E[y(t+i) | y(t-1), y(t-2), …, y(1)] or E(t)y(t+i)

Question 50

Why do we use logs for growth, interest rates etc?

Answer 50

Logging tends to convert multiplicative relationships to additive relationships, and it tends to convert exponential (compound growth) trends to linear trends. By taking logarithms of variables which are multiplicatively related and/or growing exponentially over time, we can often explain their behavior with linear models. A log transformation converts the exponential growth pattern to a linear growth pattern, and it simultaneously converts the multiplicative (proportional-variance) seasonal pattern to an additive (constant-variance) seasonal pattern.

Question 51

ARMA: What part of the equation makes it auto regressive and moving average?

Answer 51

The difference equation given by the homogeneous part of the original equation.
The Moving average part is the {x(t)} sequence.

Question 52

When would we have an ARMA (p,q) model?

Answer 52

If the homogeneous part of the difference equation contains p lags and the model for x(t) contains q lags.

Question 53

ARMA (p,q): What would we call this if p=0?

Answer 53

A pure moving-average process denoted by MA(q)

Question 54

ARMA (p,q): What would we call this if q=0?

Answer 54

A pure autoregressive process denoted by AR(p)

Question 55

What makes a time-series covariance stationary? What else can we call this?

Answer 55

if its mean and all auto covariances are unaffected y a change of the time origin. It can also be called weakly stationary.

Question 56

For a covariance-stationary series, how can we define the autocorrelation between y(t) and y(t-s)? What is an important characteristic of the gamma’s expressed?

Answer 56

𝝆(s) ≡ γ(s) / γ(0)

Where γ(0) and γ(1) are described in Brainscape Econometrics Companion 13.
γ(0) and γ(s) are both time-independent.

Question 57

Define Covariance Stationary in notation.

Answer 57

See companion 13.

Question 58

How does autocorrelation occur? (3)

Answer 58

1. Omitting important variables (having serially correlated errors): If a variable is omitted that is persistent over time, this would be incorporated into the error term 𝛍(i), so there would be persistence in 𝛍(i).
2. If we functionally miss-specify the model. This could be for instance due to us not noticing a curve in the set of data, and assuming the model is linear.
3. Measurement Error (in the independent variable): If the measurement error is persistent in 𝛍(i) and α(i), we have autocorrelation.

Question 59

What is autocorrelation?

Answer 59

The correlation of a time series, with itself, with some sort of shift. e.g. temperature over month starting at t=1, correlated with the temperature over a month starting at t=2..
When a function is autocorrelated, it will tend towards - but with a lag. The faster the ruction reaches zero, (the steeper the gradient), and the less autocorrelated the MA is, meaning the less persistent it is.

Question 60

Persistence?

Answer 60

Is expected when the time series has a shock, and the shock persists for some amount of time. This can be caused by something in the error term being persistent over time, such as when we have ommiteted a variable.