ECON0022 ECONOMETRICS FOR MACROECONOMICS AND FINANCE
Table of Contents
Lecture 2: AR models and Time Series Regressions .......................................................................................................................... 3
1. Autoregression.................................................................................................................................................................................... 3
2. Forecasts................................................................................................................................................................................................ 5
3. Stationary & Mixing .......................................................................................................................................................................... 6
4. Inference when AR is stationary ................................................................................................................................................. 9
Lecture 3: AR, ARMA, ADL models ......................................................................................................................................................... 11
1. Stationarity & Mixing – AR, MA & ARMA............................................................................................................................... 11
2. Autoregressive – Distributed Lag (ADL) Model ................................................................................................................. 15
Lecture 4: Forecasting & Lag Length Selection ................................................................................................................................. 16
1. Forecast Uncertainty ..................................................................................................................................................................... 16
2. Forecast Intervals ........................................................................................................................................................................... 17
3. Multi-period forecasts .................................................................................................................................................................. 18
4. Lag Length Selection: Information Criteria .......................................................................................................................... 20
Lecture 5: Non-stationaiy: Trends/Unit Roots ................................................................................................................................. 22
1. Trends & Unit roots ....................................................................................................................................................................... 22
2. General stochastic trends ............................................................................................................................................................ 28
Lecture 6: Asset Return Predictability ................................................................................................................................................. 30
1. Asset Prices & Returns.................................................................................................................................................................. 30
2. Stylized Facts .................................................................................................................................................................................... 31
3. Asset Return Predictability ......................................................................................................................................................... 32
4. Testing RWH ..................................................................................................................................................................................... 33
5. “Technical” Trading rules ............................................................................................................................................................ 36
6. Regression-based tests ................................................................................................................................................................. 37
Lecture 7: Structural Breaks ..................................................................................................................................................................... 38
1. About Structural Breaks............................................................................................................................................................... 38
2. Chow Test (known break dates)............................................................................................................................................... 38
3. Quandt Likelihood Ratio (QLR) Test (unknown break dates) ..................................................................................... 39
4. Pseudo-forecasting evaluation .................................................................................................................................................. 41
5. Rolling window OLS estimation................................................................................................................................................ 43
Lecture 8: Dynamic Causal Effect ........................................................................................................................................................... 44
1. Dynamic Causal Effects & Distributed Lag model ............................................................................................................. 44
2. Estimation of dynamic causal effects with exogenous regressors ............................................................................. 44
3. HAC standard errors (refer to L2) ........................................................................................................................................... 45
4. Estimation of Dynamic Causal Effects with Strictly Exogenous Regressors .......................................................... 48
, 5. Is Exogeneity plausible? ............................................................................................................................................................... 50
Lecture 9: CAPM............................................................................................................................................................................................. 51
1. Motivation .......................................................................................................................................................................................... 51
2. Mean-Variance Portfolio Selection .......................................................................................................................................... 51
3. CAPM .................................................................................................................................................................................................... 52
4. Estimation and testing CAPM → β constant over time ................................................................................................. 53
5. Estimation and testing Conditional CAPM → β vary over time................................................................................. 55
Lecture 9: Arbitrage Pricing Model (APT) .......................................................................................................................................... 57
1. APT........................................................................................................................................................................................................ 57
2. Econometrics of APT ..................................................................................................................................................................... 58
3. Some Critical Issues ....................................................................................................................................................................... 58
Lecture 10: Volatility models ................................................................................................................................................................... 60
1. Volatility ............................................................................................................................................................................................. 60
2. ARCH (1) Model ............................................................................................................................................................................... 60
3. Estimation of ARCH (1) → Gaussian MLE ............................................................................................................................ 62
4. Inference of ARCH (1) ................................................................................................................................................................... 64
5. ARCH(q) and GARCH ..................................................................................................................................................................... 65
6. Forecasting volatility ..................................................................................................................................................................... 68
Summary of all tests ..................................................................................................................................................................................... 72
,Lecture 2: AR models and Time Series Regressions
1. Autoregression
(i) AR(1)
● µ and ϕ do not have causal interpretations
● T-test
○ Φ = 0 → 𝑌𝑡−1 not useful to forecast 𝑌𝑡
○ Φ ≠ 0 → 𝑌𝑡−1 useful to forecast 𝑌𝑡
● Magnitude of Φ → degree of persistence in 𝑌𝑡
Solve the model recursively
𝑌𝑡 can be expressed as a function of
● Initial value 𝑌0
● Current & past shocks 𝜀𝑡 , 𝜀𝑡−1 , . . . 𝜀1
Estimate by OLS
(ii) AR(P)
● Coefficients do not have causal interpretations
● F-test → test if 𝑌𝑡−2 , . . . 𝑌𝑡−𝑝 useful to forecast 𝑌𝑡
● Determine lag order p using
○ F-test
○ T-test
○ “Information criterion”
,Regressors (lags) help to predict 𝑌𝑡 in a
● Statistical sense
F-test: statistical significance
● Substantive sense
Substantial increase in
* Digression 1: regress vs var command in STATA
❖ Yield exact same estimates/coefficients
❖ Regress → non-robust standard errors
Var → robust/non-robust standard errors
* Digression 2: use 𝛥 ln 𝑓𝑡 and not ln 𝑓𝑡
❖ (refer to L5)
,2. Forecasts
● 𝑌𝑇+1 | 𝑇 → Forecast of 𝑌𝑇+1 based on 𝑌𝑇, 𝑌𝑇−1 . .. using population coefficients
● 𝑌̂𝑇+1 | 𝑇 → Forecast of 𝑌𝑇+1 based on 𝑌𝑇, 𝑌𝑇−1 . .. using estimated coefficients
● Forecast error (𝑒 𝑇+1 ) =𝑌𝑇+1 | 𝑇 - 𝑌̂𝑇+1 | 𝑇
Example: For AR(1)
● 𝑌𝑇+1 | 𝑇 = 𝜇 + 𝜙𝑌𝑇
● 𝑌̂𝑇+1 | 𝑇 = 𝜇̂ + 𝜑̂𝑌𝑇
,3. Stationary & Mixing
(i) Stationarity (≠independent)
● 𝑌𝑇 is stationary if its probability distribution does not change over time
ie. Joint distribution of (𝑌𝑆+1, 𝑌𝑆+2, . . . , 𝑌𝑆+𝑇 )does not depend on S
● Jointly stationary:
𝑋𝑇 and 𝑌𝑇 are jointly stationary if the joint distribution of (𝑋𝑆+1, 𝑌𝑆+1, 𝑋𝑆+2, 𝑌𝑆+2, . . . , 𝑋𝑆+𝑇, 𝑌𝑆+𝑇 )does
not depend on S
Implies that:
1. Historical data informative about future
2. Can use historical data to estimate characteristic of {𝑌𝑇 }
3. Variances, covariances do not change over time
4. Can use standard inference tools
❖ Strictly stationary (stricter)
Implies Cov (𝑌𝑆, 𝑌𝑆+𝑘 )= Cov (𝑌𝑇, 𝑌𝑇+𝑘 ) ⇒autocorrelations are constant
❖ Covariance stationary (weaker)
a. E(𝑌𝑇 ) does not change with t
b. Cov (𝑌𝑇, 𝑌𝑇+𝑘 ) does not change with t
(ii) Mixing (dependence)
Stationary time series is mixing if
Geometrically mixing
If Cov (𝑍𝑡, 𝑍𝑡+𝑘 ) → ∞at a geometric rate
Implications:
1. Process forgets the past over time
2. 𝑍𝑡 is independent of 𝑍𝑡+𝑘 as k → ∞
,(iii) STATA: Stationary & mixing
Corrgram: Sample autocorrelation gradually converging to 0 → possibly stationary & mixing
Sample autocorrelation osciliate and go to zero
(iv) Variance adjustment
If time series is stationary & geometrically mixing, satisfies LLN & CLT:
LLN
Implications:
can estimate population moments with sample moments (consistency)
In large sample, can consistently estimate autocorrelation function
CLT
Implications:
If estimator is normally distributed, can use t-stats and F-stats
, Asymptotic variance/long-run variance
𝑇−𝑘
= Var (𝑍𝑇 ) + 2 ∑𝑇−1
𝑘=1 𝑉𝑎𝑟 (𝑍𝑇 )𝜌𝑧 (𝑘)
𝑇
𝑇−𝑘
= Var (𝑍𝑇 ) + 𝑉𝑎𝑟 (𝑍𝑇 ) 2 ∑𝑇−1
𝑘=1 𝜌𝑧 (𝑘)
𝑇
𝒇𝑻 (adjust for potential autocorrelation)
= Var (𝑍𝑇 ) ( 1 + 𝑓𝑇 )
Heteroskedasticity and Autocorrelation-Consistent (HAC) SEs
• In i.i.d we generally assume Cov (𝑍𝑡, 𝑍𝑡+𝑘 )= 0
Therefore conventional OLS SEs (heteroskedasticity robust or not) are wrong when 𝑍𝑡 is
autocorrelated (we assume 𝑓𝑇 =0)
• Robust SE only deals with heteroskedasticity
• Need HAC SEs to produce SEs that are robust to autocorrelation
• Similar to “clustered standard errors” in panel data
o Clustered SEs utilize that there are n observations at time t
o But in time series only 1 observation at time t
“Newey-West” HAC standard errors (for time series)
m
❖ Can estimate Var (𝑍𝑇 ) with ̂
𝑉𝑎𝑟 (𝑍𝑇 ) because
o Stationary
o LLN
❖ Estimate 𝜌𝑧 (𝑘) with 𝜌
̂(𝑘)
𝑧
̂ (𝑍𝑡 ,𝑍𝑡+𝑘 )
𝑐𝑜𝑣
Where 𝜌̂𝑧 (𝑘) = ̂ 𝑡)
𝑉𝑎𝑟(𝑍
❖ m is a truncation parameter
o Choose m<T
o Goldilock’s method
o S&W rule of thumb: m = 𝟎. 𝟕𝟓𝑻 𝟏/𝟑
o Wooldridge rtule of thumb: m = 𝟎. 𝟎𝟐𝟓 𝑻 𝟏/𝟒
