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ECON0022 (Econometrics for Macroeconomics and Finance) Summary - UCL Economics BSc Third Year

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Econometrics for Macro and Finance (ECON0022) (ECON0022)

Institution
University College London (UCL)

Summary of Econometrics for Macroeconomics and Financ taught in ECON0022 (Year 2022/2023) Detailed notes from lecture notes, textbooks and other materials. Topics covered include: 1) Introduction to Time Series and Autoregressive (AR) Models, 2) Time Series Asymptotic Theory and Autoregressiv...

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Econometrics for Macro and Finance (ECON0022) (ECON0022)

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UNIVERSITY COLLEGE LONDON

DEPARTMENT OF ECONOMICS
Economics BSc (Econ)
Third Year – Term 1

ECONOMETRICS FOR
MACROECONOMICS AND
FINANCE
ECON0022
Rodrigo Antón García

rodrigo.garcia.20@ucl.ac.uk

London, 2022

,
, Contents
Topic 1 – Introduction to Time Series and Autoregressive (AR) Models.

o Topic 1.1 – Introduction to Time Series Data and Serial Correlation. 1
o Topic 1.2 – Casual Inference Versus Prediction. 3
o Topic 1.3 – The AR(1) Model: Estimation and Forecasting. 4
o Topic 1.4 – The AR(p) Model. 8

Topic 2 – Time Series Asymptotic Theory and Autoregressive (AR) Models.

o Topic 2.1 – Limit Theory for Stationary and Mixing Time Series. 11
o Topic 2.2 – The Autocorrelation Function. 15
o Topic 2.3 – Time Series Regression. 16
o Topic 2.4 – OLS Asymptotic Theory for Stationary AR Models. 18

Topic 3 – Univariate Time Series Models: Autoregressive (AR), Moving Average
(MA), ARMA and Autoregressive Distributed Lag (ADL) Models.

o Topic 3.1 – Moving Average (MA) Models. 25
o Topic 3.2 – Autoregressive (AR) Models. 30
o Topic 3.3 – ARMA Models. 41
o Topic 3.4 – Autoregressive Distributed Lag (ADL) Models. 41
o Topic 3.5 – Lag Length Selection (AR and ADL Models). 43
o Topic 3.6 – Diagnostic Tests. 45

Topic 4 – Forecasting and Model Selection.

o Topic 4.1 – Principles of Forecasting. 47
o Topic 4.2 – Forecasting with ADL Models. 49
o Topic 4.3 – The MSFE Terms and Dimension-Reduction Techniques. 52
o Topic 4.3 – Granger Causality. 54
o Topic 4.4 – Forecast Evaluation. 54
o Topic 4.5 – Multi-Step Forecasting. 57

Topic 5 – Trends and Cycles.

o Topic 5.1 – Deterministic Trends. 60
o Topic 5.2 – Stochastic Trends and Unit Roots. 63
o Topic 5.3 – The Beveridge-Nelson Decomposition. 73
o Topic 5.4 – The Hodrick-Prescott Filter. 64

,Topic 6 – Asset Return Predictability.

o Topic 6.1 – Asset Prices and Returns. 75
o Topic 6.2 – Stylized Facts About Returns. 76
o Topic 6.3 – The Efficient Market Hypothesis (EMH). 78
o Topic 6.4 – Asset Return Predictability. 79

Topic 7 – Structural Breaks.

o Topic 7.1 – Structural Breaks. 89
o Topic 7.2 – Consequences of Structural Breaks. 90
o Topic 7.3 – Detecting Structural Breaks. 92

Topic 8 – Dynamic Causal Effects.

o Topic 8.1 – Dynamic Casual Effects. 97
o Topic 8.2 – Dynamics Casual Effects in Distributed Lag Models. 97
o Topic 8.3 – Inference in Distributed Lag Models. 99
o Topic 8.4 – HAC Estimators. 100

Topic 9 – The Capital Asset Pricing Model (CAPM) and Factor Models (APT).

o Topic 9.1 – Mean-Variance Portfolio Selection. 103
o Topic 9.2 – The Capital Asset Pricing Model (CAMP). 106
o Topic 9.3 – Implementation and Testing of the CAPM. 108
o Topic 9.4 – Problems of the CAPM. 112
o Topic 9.5 – The Arbitrage Pricing Theory (APT). 112
o Topic 9.5 – Data Snooping and Unstable Betas. 115

Topic 10 – Volatility Models.

o Topic 10.1 – Conditional Volatility. 116
o Topic 10.2 – Realized Volatility Estimators. 118
o Topic 10.3 – ARCH and GARCH Models. 119
o Topic 10.4 – Estimation of ARCH and GARCH Models. 123
o Topic 10.5 – Inference in ARCH and GARCH Models. 126
o Topic 10.6 – Volatility Forecasting. 127

,ECON0022 – Econometrics for Macroeconomics and Finance Rodrigo Antón García

ECON0022: ECONOMETRICS FOR MACROECONOMICS AND FINANCE

Topic 1 – Introduction to Time Series and AR Models.

o Topic 1.1 – Introduction to Time Series Data and Serial Correlation.

Time Series Data sets consists of observations on a variable or several variables over
time. Data is collected on the same observational unit at multiple time periods.

The same unit is observed at different points in time.

Some of the uses of time series data are:

- Forecasting. E.g., What will next year’s unemployment level be?
- Estimation of dynamic causal effects. E.g., If the Fed increases the Federal Funds
rate now, what will be the effect on the rates of inflation and unemployment in 3 months?
- Estimation of structural dynamic (macro) models.

Relative to cross-sectional data and models, time series raises new technical issues
such as time lags, correlation over time (serial correlation or autocorrelation), standard
error calculation when they serially correlated, persistent data, latent variables dynamics.

The major difference between cross-sectional data and time series data is the existence
of correlation between periods, past data displays information about the present data,
and as a result observations are no longer independently and identically distributed (i.i.d.)

• Time Series Basics: Notation.

We are interested in modelling a random process ! that we observe over multiple time
periods "! . That is, !" = value of ! in period " = 1, 2, 3, etc. Thus, our data set is defined
as {!# , … , !$ }, and there are , ≥ 1 observations on the time series variable !, where the
time index is relative to the frequency of the observations, e.g., monthly, quarterly, etc.

Note we will only consider consecutive, evenly-spaced observations (for example,
monthly, 1960 to 1999, no missing months). This is because, missing and unevenly
spaced data introduce technical complications. Dealing with these require more
advanced techniques that go beyond the contents of this module.

• Time Lags, First Differences (FD) and Growth Rates.

Instead of working with the raw data directly, very often we manipulate data to obtain
variables of interest such as lags and first difference from which regression can be build.

- The first time lag of a time series !" is given by !"%# .
- The first difference of a time series, ∆!" , is the change between periods " − 1 and
" and it is given by ∆!" = !" − !"%# . The first difference of the natural logarithm of ! is
∆ log(!" ) = log(!" ) − log(!"%# ). Note log(… ) usually refers to the natural logarithm ln(… ).

1

,ECON0022 – Econometrics for Macroeconomics and Finance Rodrigo Antón García

Therefore, the percentage change of a time series !" between periods " − 1 and " is,

%∆!" = 100(!" − !"%# )/!"%# ≈ 100∆ log(!" )

Of course, the approximation is most accurate when the percentage change is small.

All of the above can be thought of as new time series derived from the original one. For
instance, it is helpful to visualize an autoregressive model :;(<) , where we are
regressing the original time series on < lagged time series shifted < periods behind.

If we consider the quarterly rate of inflation at an annual rate, for example the US
Consumer Price Index (CPI), the CPI in Q1 of 2004 is 186.57 CPI whereas that in Q2 of
2004 it is 188.60. Thus, the exact percentage change in CPI between these periods is
then % ∆BCD&'#(%)& = 100(!" − !"%# )/!"%# = 100(188.60 − 186.57)/186.57 = 1.088%.

At an annual rate this is, 4 × 1.088% = 4.359%, following the logarithmic approximation,

%∆!" ≈ 100∆ log(!" ) 4 × 100[log(188.60) − log(185.57)] = 4.329%

• Serial Correlation (Autocorrelation).

The correlation of a series with its own lagged values is called autocorrelation. In terms
of time lags, for the variances JKL(!" ) and JKL(!"%# ), the first population autocorrelation
of !" is BMLL(!" , !"%# ), and the first population autocovariance of !" is BMN(!" , !"%# ).

BMN(!" , !"%# ) R*! ,*!"#
BMLL(!" , !"%# ) = → Q*! ,*!"# =
OJKL(!" )JKL(!"%# ) R*! R*!"#

These are population moments: they describe the population joint distribution of (!" , !"%# ).

Having defined the first order autocorrelation between !" and !"%# , we can think of
higher-order autocorrelation when we increase the amount of time lags in the regression.

The Sth-order autocorrelation of !" is the correlation between !" and its Sth lag !"%, . And
so, the Sth-order autocovariance of !" is the covariance between !" and its Sth lag !"%, .

BMNT!" , !"%, U R*! ,*!"$
BMLLT!" , !"%, U = → Q*! ,*!"$ =
R*! R*!"$
VJKL(!" )JKLT!"%, U

The Sth sample autocorrelation is an estimate of the Sth population autocorrelation:

+ %&! , &!"# (
!"* + %&! , &!"# (
!"* 5 9 )%7'"( − 7
4$!,$!"# ∑)'*(+,(7' − 7 9(
$ %&! , &!"# ( =
!"## = → 3
4$!,$!"# = =
+ (&! )-.#
,-.# + (&! ) + (&! )
-.# 4 &$
5 9)&
∑)'*,(7' − 7
!

W T!" , !"%, U = # ∑$"-,.#(!" − !Y)T!"%, − !YU and JKL
where BMN W (!" ) = # ∑$"-#(!" − !Y)& .
$ $

2

,ECON0022 – Econometrics for Macroeconomics and Finance Rodrigo Antón García

Note that for the sample autocorrelation, the summation is over " = S + 1 to ,, and that
the divisor is the variance of !$ . This is the conventional definition used for time series.

W (!" )
The reason why the divisor in the sample autocorrelation function is the variance JKL
is because the standard deviation of a stationary time series is the same at all times, and
since we are assuming stationarity, JKL W (!" ) = JKL
W T!"%, U, and hence the denominator
W (!" ), where " is taking by default since all variances are equal.
simplifies to JKL

o Topic 1.2 – Casual Inference Versus Prediction.

In ECON0019, we covered how to use regression models for causal inference. For
example, if we consider ! = [' + [# \ + ], where ^[]|\] = 0, the casual effects of \ are
given by [# = `^[!|\]/`\, . The OLS estimator of [# only has a meaningful interpretation
if the conditional-mean dependence assumption ^[]|\] = 0 holds. Thus, if ! is a non-
linear function of \ and/or ^[]|\] = 0 fails then OLS is not useful for causal inference.

Now, suppose instead that you simply want to predict ! for some datapoint out of sample,
i.e., you want to make a prediction or a forecast. For example, say you want to predict
!/.# = person’s income level when you only know the person’s \/.# = years of
education. Then, we would expect that the relationship between ! and \ might be
nonlinear, and moreover \ is most likely to correlate with other factors affecting income.

Suppose you still decide to predict !/.# by !/.# = [a' + [a# \/.# Is this a useful prediction?

Yes, it is! The idea behind OLS is that it gives you the best linear prediction regardless
of actual Data Generating Process (DGP). Thus, once the sample OLS is computed to
target the population OLS, our prediction is in a sense, the best prediction we could make.

Of course, we know that if ! is actually a non-linear function of \ then this prediction
might still be poor, since it comes with a larger prediction error. Therefore, it is still true
that we still care about finding a good model, but overall, we are less concerned with the
model being completely correct. After all, it is never completely accurate anyway.

Here, OLS is our selection of specification and our focus is not in a casual relationship.
In forecasting, we are not concerned with misspecification anymore but instead on other
factors like large prediction errors and creating models that derive precise predictions.

This means that bias (OBV) is no longer a concern for prediction: the [a, ’s are no longer
defined in terms of the model itself and the [a, ’s do not carry a casual interpretation. The
[a, carry the value of the slope coefficient that minimizes the prediction error. However,
note that we do care about OBV in terms of leaving useful predictors outside the model.

bMLcdKe"fgh → ifgfjfkc CLclfd"fMg ^LLML (mM BKenKo Dg"cL<Lc"K"fMg)

The coefficients are just numbers that are useful for accurate prediction of our factor.
Importantly, this discussion in a cross-sectional setting also holds true in a time series.

3

, ECON0022 – Econometrics for Macroeconomics and Finance Rodrigo Antón García

o Topic 1.3 – The AR(1) Model: Estimation and Forecasting.

We now wish to build models to carry out prediction in a time series context (forecasting).
To be more precise, it is the prediction in a time series setting that is called forecasting.

A natural starting point is to use past values of ! (that is, !"%# , !"%& ,…) to forecast !"
when autocorrelation is present.

An autoregression is a regression model in which !" is regressed against its own lagged
values. The number of lags used as regressors is called the order of the autoregression.

- In a first order autoregression, !" is regressed against !"%# .
- In a p-th order autoregression, !" is regressed against !"%# , !"%& ,…, !"%0 .

• The First Order Autoregressive (AR(1)) Model.

The population first-order autoregressive model AR(1) regresses the first lag of !" ,

!" = p + q!"%# + ]" , " = 2,3, …

where q is the slope coefficient, !"%# is the first time lag of !" and ]" are random
unobserved factors that affect !"%# . Note that the notation we will use changes slightly.

Remember, this model is used for forecasting and not for casual inference, thus q does
not have causal interpretation. In forecasting, q pins down the precise dynamics of !" ,
and, as a result, the interpretation of q is capturing the implied dynamics of the AR model.

Testing the null hypothesis r1 : q = 0 against the alternative r# : q ≠ 0 provides a test
of whether the first lag is not useful for forecasting !" . The magnitude of q determines
the degree of persistence in !" , the larger q, the stronger the autocorrelation.

The error term ]" is sometimes also called the shock as it captures surprise movements
in this period’s value of ! once we have controlled for past observations (!"%# ). Since we
do not wish to give a causal interpretation to the model, ]" is not required to be mean-
independent of past values of !, but if it is, our forecast will be optimal in a certain sense.

- Interpretation of the AR(1) Model.

The AR(1) model is dynamic since relates past values to current ones. Thus, we can
"solve" the model recursively so as to express the current value of !" as a function of
current and past shocks.

!& = p + q!# + ]&
!2 = p + q!& + ]2 = p + q(p + q!# + ]& ) + ]2 = p + qp + q & !# + q]& + ]2
⋮
"%& "%&
!" = p + q!"%# + ]" = p v q ! + q "%# !# + v q ! ]"%!
!-' !-'

4

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