Nonstationarity, Autocorrelation, and Collinearity in Modeling Forest Products Marketsode g o est oducts a ets

Nianfu Song and Sun Joseph ChangSchool of Renewable Natural Resources

Louisiana State Universityy

ObjectivesObjectives

• Show the existence of nonstationarityShow the existence of nonstationarity• Consequences of nonstationarity and

problems in current studiesproblems in current studies• Estimation Methods with nonstationarity

d tdata.• Long-run and short-run• Re-explanation of some past research.

Estimation MethodsEstimation Methods• 2SLS, 3SLS

Adams and Haynes 1996;– Adams and Haynes, 1996;– Adams et al. 1986; – Baek and Yin, 2006;

Bernard et al 1997 (NL3SLS);– Bernard et al. 1997 (NL3SLS);– Latta and Adams, 2000 (NL3SLS);– Lewandrowski et al. 1994.

• OLS• OLS– Rockel and Buongiorno, 1982, Adams et al. 1992, Cardellichio,

1990

• ML– Rao et al. 2004; Myneni, 1994

• Kalman Filter Adams et al 1992• Kalman Filter Adams et al. 1992

Nonstationarity, Unit Root

• Stationarity is a basic requirement for Central Limit Theorem (time series used in lumber models are mostly nonstationary)Theorem (time series used in lumber models are mostly nonstationary).

• OLS with nonstationary {zt} is spurious when there is no cointegrationis no cointegration

• Weakly stationary: E[zt] and Var[zt] are constant, and Cov[zt, zt k] is finite and independent of tCov[ t, t-k] s e d depe de o

• If A(L)∆zt = εt, the time series process {zt} has a unit root. With unit root, {zt} is nonstationary. , { t} y

Softwood Lumber Prices are N iNonstationary

600

500

600

BF

200

250

ndex

300

400

rric

esin

$/M

B

150

mbe

rPri

ceI

100

200

Lum

ber

50

100

onth

lyLu

m

0

1990

1990

1991

1992

1993

1993

1994

1995

1996

1996

1997

1998

1999

1999

2000

2001

2002

2002

2003

2004

P1 P2 P3 P4

094

795

095

395

796

096

496

797

197

497

898

198

598

899

299

599

800

200

5

M

P1 P2 P3 P4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2

Softwood Timber Price IndexSoftwood Timber Price IndexSoftwood timber price index

250

300

200

250

100

150

0

50

2 3 4 6 7 8 0 1 3 4 5 7 8 0 1 2 4 5

1982

1983

1984

1986

1987

1988

1990

1991

1993

1994

1995

1997

1998

2000

2001

2002

2004

2005

Housing Starts, Mortgage Rate and DPIHousing Starts, Mortgage Rate and DPI

9000) 20

7000

8000

9000

on o

f 200

0$)

1618

20

4000

5000

6000

0), D

PI(B

illio

10

1214

age

Rate

2000

3000

4000

g S

tart

s(10

00

4

68

Mor

ga

0

1000

71 74 77 80 83 86 89 92 95 98 01 04

Hou

sin g

0

24

Apr

-7

Apr

-7

Apr

-7

Apr

-8

Apr

-8

Apr

-8

Apr

-8

Apr

-9

Apr

-9

Apr

-9

Apr

-0

Apr

-0

Nonstationary Time Series Studies FoundNonstationary Time Series Studies Found

• The “law of one price” studies withThe law of one price studies with cointegration tests

--Jung and Doroodian (1994), Nanang (2001)g ( ), g ( )--Buongiorno and Uusivuori (1992), Alavalapati et al. (1997) and Hänninen (1998), Yin et al. (2002), Yin et al (2005)et al. (2005).

• ECM models (MLE)--Restricted ECM Toppinen (1998) for Finnish--Restricted ECM, Toppinen (1998) for Finnish sawlog--Baek and Yin (2006)

Cointegration

• When a linear combination of nonstationary variables yis stationary, these variables are cointegrated.

• Cointegration exists as long as market equilibrium iexists.

• OLS with nonstationary {zt} is “super consistent” when there is cointegrationwhen there is cointegration.

• Cointegration has been used in testing “the Law of One Price” by many in forestry literature and in at O e ce by y o es y e u e dleast one structural models for timber, and one for lumber.

Autoregressive RepresentationAutoregressive Representationqp

∑∑• Or

t1j

jtj0i

itit yxy εφβ ++= ∑∑=

−=

−

Or exists)L(wherex)L(By)L( 1

ttt−=+ ΓεΓ

zxyorzx)1(By)1(relationrunlongWith

+=+=−

αΓ tttttt zxyorzx)1(By)1( +=+= αΓ

Estimate the Autoregressive Model i h 2SLSwith 2SLS

• The coefficients of the autoregressiveThe coefficients of the autoregressive representation can be estimated with 2SLS or 3SLS2SLS or 3SLS.

• The invertability and cointegration of the model must be checkedmodel must be checked.

• Long-run relation can be obtained by t f i th ti t d t itransforming the estimated autoregressive representation of the cointegration.

Triangle RepresentationTriangle Representation

• The source of nonstationarity of endogenousThe source of nonstationarity of endogenous variables is the unit roots of the exogenous variables.

ttt zxy α +=

ttx εδ∆ +=

Estimate the Coingretion Relations i h OLS 2SLSwith OLS or 2SLS

• The OLS is supper consistentThe OLS is supper consistent. • 2SLS is consistent.

With t l ti th t i i• With autocorrelation the true variance is large. However the true variance some ti i t bt i d b fttimes is not obtained by software.

ECM RepresentationECM Representation

1p1p −−

t1t

p

1jjti

p

0iitit uzyxy εγ∆α∆β∆ ++++= −

=−

=− ∑∑

Estimate Structural ECM with MLEEstimate Structural ECM with MLE

• With restrictions an ECM can be estimatedWith restrictions an ECM can be estimated by MLE. However the cointegration test with no restriction must be applied first, ppand the restrictions have to be tested by a ki-square test.

• It is a pure statistical method without any knowledge prior to estimating the

i t ticointegration.• An example is Toppinen (1998).

Misspecificationp

• With cointegration DynamicWith cointegration, Dynamic autoregressive model (ECM equivalent) is the data generating function and thethe data generating function, and the models of differences (ARIMA) are misspecified (Engle and Granger 1987)misspecified (Engle and Granger, 1987).

• By this rule, at least one of the lumber structure models published recently isstructure models published recently is misspecified.

Invertible and Stationary C ffi i M iCoefficient Matrix

• The roots of the absolute value of the matrix Г(L)The roots of the absolute value of the matrix Г(L) equal to zero must be outside the unit circle– In the case of one lag, the coefficient of Y,t-1 or the ratio of the

ffi i f d h b l h d hcoefficient of Y,t-1 and Y,t have to be less than 1 and greater than -1.

Thi diti b i l t d b h ki th• This condition can be implemented by checking the eigenvalues of a matrix F (Hamilton, 1994, page 259) The absolute values of the eigenvalues must259). The absolute values of the eigenvalues must be smaller than unit to ensure the stationarity of the data generation system.of the data generation system.

Past StudiesPast Studies

• The cointegration condition was satisfiedThe cointegration condition was satisfied for models estimated with OLS, 2SLS or 3SLS because of the routine reports of the3SLS because of the routine reports of the DW statistics.

• Unknown for other methods• Unknown for other methods.• Some times the invertability condition is

t ti fi dnot satisfied.

Coefficient Matrix of a Recently Published Lumber Model

-0.35

1.91 1.78 1.70

0.25 -0.36

-1.61 -1.45 -1.39

0.55 -.49 -0.41

-0.16

-0 61-0.61

-0.30

-0.14 0.17 0.29

The Eigenvalues of the F MatrixThe Eigenvalues of the F Matrix

• 0.962813

• 0.508607• 0.508607• 0.026072• 0.026072• -0.060270.06027• -0.06027• -0.19521• -0.19521• -0.48036• -0 480360.48036

• -1.3205 (> 1, showing nonstationarity)

Similar problem in another paperSimilar problem in another paper

• With AR(1) when ρ>1 the expected variance• With AR(1), when ρ>1 the expected variance is infinite

• Values of ρ in the paper• Values of ρ in the paper0.80, 0.76, 0.89, 1.021 25 0 76 0 99 1 961.25, 0.76, 0.99, 1.960.44, 1.02, 1.03, 0.98

Implication of Frequencyp q y• Estimates with monthly prices approximately

equals that with annual prices in the long-run, q p g ,• But the short-run effect implied by the ECM is

different

Observations over TimeObservations over Time

Data

54

56

58

60

62

dex

3334353637

lum

e

44

46

48

50

52

54

Pric

e in

d

272829303132

Out

put V

o

445 6 7 8

year

27

A one time increase in price (small dots over time)A one time increase in price (small dots over time)

The observed outputs (large diamonds over time)

With NO LagsWith NO Lags

• Pink line: regressedPink line: regressed curve with OLS

• Diamonds: output Regression

37pagainst price.

33

34

35

36

37

olum

e29

30

31

32

48 50 52 54 56 58 60 62Vo

48 50 52 54 56 58 60 62

Price

Pink line: regressed curve with OLS

Diamonds: output against price with data on the left graph .

Short-Run EffectShort Run Effect

• The coefficients of differences of a ECM h th t

Regression

36

37

shows the current response.

32

33

34

35

Volu

me

29

30

31

48 50 52 54 56 58 60 62

Price

With Lag VariablesWith Lag Variables

RegressionRegression

36

37

33

34

35

lum

e

30

31

32Vol

29

30

48 50 52 54 56 58 60 62

Price

The Long-Run EffectThe Long Run EffectRegression

36

37

33

34

35

olum

e

30

31

32Vo

2948 50 52 54 56 58 60 62

Price

Lumber Demand ElasticityLumber Demand Elasticity

• The reported elasticities in forest literatureThe reported elasticities in forest literature are mostly short-run elasticity in one period of time (lags included). p ( g )

• or ones below their true numbers when the autocorrelation exists in the error terms.

• According to our recent research, the long-run elasticity could be -0.71 when the yoften referred -0.17 is transformed according to the original estimated model.

ConclusionConclusion

• The long-run elasticity may be greaterThe long run elasticity may be greater than what we thought.

• Some past models were misspecified (the• Some past models were misspecified (the impact of the misspecification is unknown)S ti t d t t ti• Some estimated system are nonstationary or unstable.

•

SuggestionsSuggestions

• With level data if the errors are wellWith level data, if the errors are well behaved, the unit roots of the time series can be ignoredcan be ignored.

• The stationarity of a estimated time series model with lags of endogenous variablesmodel with lags of endogenous variables or autoregressive errors should always be checkedchecked.