Authors: Péter Elek, Anikó Bíró Supervised by: Péter Elek

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ECONOMETRICS

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ECONOMETRICS

Sponsored by a Grant TÁMOP-4.1.2-08/2/A/KMR-2009-0041 Course Material Developed by Department of Economics,

Faculty of Social Sciences, Eötvös Loránd University Budapest (ELTE) Department of Economics, Eötvös Loránd University Budapest

Institute of Economics, Hungarian Academy of Sciences Balassi Kiadó, Budapest

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ECONOMETRICS

Authors: Péter Elek, Anikó Bíró Supervised by: Péter Elek

June 2010

ELTE Faculty of Social Sciences, Department of Economics

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ECONOMETRICS

Week 11.

Nonstationary time series

Péter Elek, Anikó Bíró

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Content

Testing nonstationarity: unit root tests Trends and seasonal components

Material: M 613 –617., 301–306., 597–602.

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Example: estimating the parameter of a random walk the limit distribution of the t-statistic

is not t-distribution!

∆X_t = c + 0·X_t–1+ _t

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Testing nonstationarity:

Dickey–Fuller test

Y_t= αY_t–1+_t

Equivalent: ∆Y_t= (α – 1)Y_t–1+ _t H₀: α = 1, H₁: α<1

Test: the usual t-statistic,

Under H₀: the so-called Dickey–Fuller-distribution Asymptotic critical values:

5%: –1,95 (t-critical value: –1,65) 1%: –2,58 (t-critical value: –2,33)

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Versions of Dickey–Fuller-test

AR(1) + constant

Y_t= c + αY_t–1+ _t

Asympt. critical value: –2,86 (5%), –3,43 (1%)

AR(1) + constant + trend

Y_t= c + δt + αY_t–1+ _t

Asympt. critical value: –3,41 (5%), –3,96 (1%)

Augmented DF test:

Y_t= c + δt + αY_t–1+ ₁·∆Y_t–1+ ₂·∆Y_t–2+…+ _k·∆Y_t–k+ _t

There are other stationarity tests as well (eg. KPSS)

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Example: is USA GDP difference or trend stationary?

Are the effects of shocks persistent or temporary?

Supply side: random walk (technological shocks)

Demand side: trend stationary

Which shocks dominate?

2,000 3,000 4,000 5,000 6,000 7,000

1960 1965 1970 1975 1980 1985 1990 1995

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Unit root test for the GDP series

The hypothesis of unit root cannot be rejected.

The conclusions are similar on larger

samples, but final

decision in the debate cannot be made.

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Why should we bother about stationarity?

Spurious trend in time series

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Spurious trend in time series II

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Spurious regression in time series

Two independent random walks:

X_t= X_t–1+ _1t Y_t= Y_t–1+ _2t

Regression: Y_t= c + βX _t+ u_t β = 0 because of

independence, but the t-test is significant!

The t-statistic does not have a limit distribution

Reason: u_t is nonstationary!

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Fitting trends in the trend stationary and the difference stationary case

Simplest trend stationary case y_t = β₀+ β₁t + u_t, u_t~ IN

Trend fitting by OLS is consistent and in this case

efficient (becaues of the independence of the error term) Differentiation also yields consistency, but the

independence of the error term does not hold any more:

y_t = β₁+ u_t– u_t–1

Difference stationary case y_t = y_t–1+ β₁+ u_t, u_t ~ IN

Trend fitting by OLS is inconsistent!

Differentiation yields a consistent (and in this case efficient) estimate: y_t = β₁+ u_t

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Hodrick–Prescott filter

y_t: original time series s_t: filtered time series

If λ = 0 then y_t= s_t for all t

If λ = ∞ then the linear trend is obtained.

A possible choice for λ: λ = 1600*(i/4)², where i is the frequency

Annual data: λ = 100 Quarterly data: λ = 1600 Monthly data: λ = 14400

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Seasonality

Two types of seasonality

deterministic (can be filtered out by using dummy variables)

stochastic (can be filtered out by taking seasonal differences)

In practice: more difficult filtering methods (e.g. TRAMO-SEATS)

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Example: daily water discharge

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ACF

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Example: wages

60000 80000 100000 120000 140000 160000 180000 200000

00 01 02 03 04 05 06 07

közszféra bérei (Ft/hó)

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Example: seasonality in the quarterly growth rate of private sector wages

ACF of residuals

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Seminar

Nonstationary time series

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Examples I.: analysis of the import time series of barium-clorid

Choice between trend stationarity and difference stationarity

by the inspection of ACF and by a formal test

Fitting a linear trend and filtering by HP-filter Fitting an AR(1) + trend to the original, and an ARMA(1,1) to the differentiated time series

Testing the uncorrelatedness of the residuals

Forecasting from the model

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Examples II

Analysis of quarterly macro time series

Fitting a deterministic seasonal component + AR(1) term, and seasonal component

+ ARMA(1,1) term Forecasting