Stochastic Trends & Economic Fluctuations King, Plosser, Stock & Watson (AER, 1991) Cesar E. Tamayo Econ508 - Economics - Rutgers November 14, 2011 Cesar E. Tamayo Stochastic Trends & Economic Fluctuations 1 / 14
Program I What they do (brief) I Econometric framework I I Cesar E. Tamayo Stochastic Trends & Economic Fluctuations 2 / 14
In a nutshell I Is a common stochastic trend (i.e. the cumulative e ect of permanent shocks to TFP) responsible for the bulk of economic uctuations? I Key questions: 1. Are the properties of post-war U.S. data consistent with BGP? 2. How much of the cyclical variation can be attributed to (innovations in) a common stochastic trend? 3. Do nominal variables play a role (as an alternative to RBC)? I Use integration and cointegration techniques to arrive at the... I Key answers: 1. Yes. 2. It all depends. About 60%-75% in a three variable model c, i, y. About 35%-44% in a model with p, R and m. 3. No. Cesar E. Tamayo Stochastic Trends & Economic Fluctuations 3 / 14
Enviroment Econometric implications Identi cation strategy Enviroment I Consider: Y t = λ t Kt 1 I Where λ t follows a log-random walk: α Nt α log λ t = µ + log λ t 1 + ζ t with : ζ t iid 0, σ 2 µ : average TFP growth I In the deterministic case (ε t = 0 8t) the per-capita aggregates C /N, Y /N and I /N will grow at µ/α along the BGP (see handout) I In the stochastic case,these aggregates will grow at (µ + ζ t )/α along the BGP. Cesar E. Tamayo Stochastic Trends & Economic Fluctuations 4 / 14
Enviroment Econometric implications Identi cation strategy Econometric implications I Testable hypothesis: what KPSW call the "great rations" C t /Y t and I t /Y t will be stationary stochastic processes. I That is (with lower case as log), X t = y t c t i t 0 I (1) but any x t y t I (0) for x = c, i. I Equivalently β 1 = 1 1 0 0 and β2 = 1 0 1 0 are such that β 0 i X t I (0). I If nominal variables are included, (at least) two additional cointegration relationships emerge: m t p t = ϑy t + γr t + ν t R t = r t + E t p t+1 I Real balances, output and interest rates are I (1) but if ν t I (0) they are cointegrated. Additionally, if R t and p t+1 are I (1) but r t I (0) then these are also cointegrated. Cesar E. Tamayo Stochastic Trends & Economic Fluctuations 5 / 14
Enviroment Econometric implications Identi cation strategy Econometric implications I X t is assumed assumed I (1) and to have the VMA representation: X t = µ + C (L) ε t (1) ε t (0, Σ ε ) serially uncorr. I Equation (1) is a reduced form version of the model. But we really care about the structural model that gives rise to (1), say: X t = µ + Γ (L) η t (2) η t 0, Σ η serially uncorr. ε t = Γ 0 η t (3) I The identi cation problem: how can we deduce the structural disturbances η t and the lag polynomial matrix Γ (L) starting from the reduced-form C (L) and ε t? Cesar E. Tamayo Stochastic Trends & Economic Fluctuations 6 / 14
Enviroment Econometric implications Identi cation strategy Identi cation strategy I Consider the small model: 3 variables and 2 CVs ) one permanent innovation, η 1 t (η2 t and η3 t have only transitory e ects). I η 1 t corresponds to ζ t in the neoclassical growth model above. I Two identi cation restrictions: 1. η 1 t is uncorrelated to (η2 t,η3 t ) (i.e., we can consider the dynamic impact of an isolated shock to η 1 t ). 2. The cointegration restrictions imply that the matrix of long-run multipliers Γ (1) (see handout for details) is: 2 Γ (1) = 4 1 0 0 1 0 0 1 0 0 i.e., j=0 γj i1 = 1 and j=0 γj i2 = j=0 γj i3 = 0 for i = 1, 2, 3. This is crucial for identi cation since (1) - (2) ) Γ (1) = C (1) Γ 0. I Augmenting the model with nominal variables requires more restrictions. 3 5 Cesar E. Tamayo Stochastic Trends & Economic Fluctuations 7 / 14
Unit root and cointegration tests IRF and FEDV Adding nominal variables Unit root tests I In the model with three variables. KPSW show (in the NBER-WP version) that y, c, i are I (1). I Moreover, c t y t and i t y t (the "great ratios") are stationary: (y, c) and (y, i) are cointegrated. source: St. Louis Fed Cesar E. Tamayo Stochastic Trends & Economic Fluctuations 8 / 14
Unit root and cointegration tests IRF and FEDV Adding nominal variables Speci cation I In order to conduct multivariate unit root and cointegration tests KPSW estimate the VECM(5): X t = δ + αβ 0 X t 5 1 + Θ j X t j + ε t (4) j=1 I Where α as the "loading" matrix and β as the cointegration matrix. I The hypothesized cointegration matrix is: β 0 1 1 0 = 1 0 1 I They also use (4) to investigate the transitional dynamics after a productivity shock (IFR and FEVD) (key question #2) Cesar E. Tamayo Stochastic Trends & Economic Fluctuations 9 / 14
Unit root and cointegration tests IRF and FEDV Adding nominal variables Unit root and cointegration tests I Multivariate unit root tests from VAR(6) in levels (VECM(5)): Multivariate Unit Root Tests (linear detrended series) stat value p-value null/alternative Johansen 35.4 0.13 0 CV / more than 0 CV Stock-Watson -29.4 0.21 3 unit roots / at most 2 unit roots Stock-Watson -29.4 <0.01 3 unit roots / at most 1 unit roots I Estimated CVs: Estimated CVs (Dynamic OLS, Stock-Watson (1989)) var β 1 β 2 ˆβ 1 ˆβ 2 c 1 0 1.00 0.00 i 0 1 0.00 1.00 y -1-1 -1.058-1.004 Wald test of BGP restrictions: χ 2 (2) =4.96 (0.08) I Answer to key question #1) yes, 9 BGP! Cesar E. Tamayo Stochastic Trends & Economic Fluctuations 10 / 14
Unit root and cointegration tests IRF and FEDV Adding nominal variables IRF and FEDV FEVD: Fraction of the forecast-error variance attributed to the real permanent shock Horizon! 1 4 8 12 24 y 0.45 0.58 0.68 0.73 0.81 1.00 (0.25) (0.27) (0.22) (0.19) (0.16) c 0.88 0.89 0.83 0.83 0.89 1.00 (0.21) (0.19) (0.18) (0.18) (0.13) i 0.12 0.31 0.40 0.43 0.47 1.00 (0.18) (0.23) (0.18) (0.17) (0.16) I This is the rst part of the answer to key question # 2: in a three (real) variable model, a good deal of cyclical variation is explained by shocks to the real permanent component. Cesar E. Tamayo Stochastic Trends & Economic Fluctuations 11 / 14
Unit root and cointegration tests IRF and FEDV Adding nominal variables Adding nominal variables I For the model with nominal variables, KPSW argue that (c y), (i y) may exhibit permanent shifts resulting from shifts in (R p). So three cointegration relations: two (real interest rate adjusted) BGP relations and a long-run money demand relation. I The results from this six variable system are used to answer part II of key question # 2: Table 4. FEVD: Fraction of the forecast-error variance attributed to the real permanent shock (six-variable model) Horizon! 1 4 8 12 24 y 0.00 0.05 0.22 0.44 0.62 1.00 (0.13) (0.14) (0.13) (0.14) (0.14) c 0.02 0.15 0.31 0.48 0.65 0.92 (0.09) (0.13) (0.18) (0.21) (0.17) i 0.11 0.06 0.14 0.27 0.33 0.97 (0.16) (0.11) (0.11) (0.16) (0.16) Cesar E. Tamayo Stochastic Trends & Economic Fluctuations 12 / 14
and comments I The identi cation strategy presented here is closely related to that used by Blanchard and Quah (1989) (long-run type). I But KPSW is a nice example of how useful cointegration relations can be in obtaining identi cation of VAR systems (more examples in Lutkepohl (2005a)). I Notably, KPSW do not test for structural breaks which in turn may invalidate their inference (Lütkepohl, 2005b). I Lütkepohl and Wolters (2003) suggest that cointegration tests may loose power if these breaks/shifts are ignored. I Att eld and Temple (2006) show that allowing for breaks in the "great ratios" the cointegration tests lend stronger support to the BGP hypothesis and thereby to the identifying restrictions used here. Cesar E. Tamayo Stochastic Trends & Economic Fluctuations 13 / 14
References Att eld, C. and J. Temple, 2006, "Balanced Growth and the Great Ratios: New Evidence for the US and UK" Centre for Growth and Business Cycle Research Discussion Paper No. 074. Blanchard, O. and D. Quah, 1989, "The Dynamic E ects of Aggregate Demand and Supply Disturbances",.American Economic Review Vol. 79, No. 4 (Sep.), pp. 655-673. King, R., C. Plosser, J. Stock and M. Watson, 1991, "Stochastic Trends and Macroeconomic Fluctuations", American Economic Review Vol. 81, No. 4 (Sep.), pp. 819-840. Lütkepohl, H., 2005a, "Structural Vector Autoregressive Analysis for Cointegrated Variables", EUI Working Paper ECO2005/2. Lütkepohl, H., 2005b, New Introduction to Multiple Time Series Analysis, Springer. Lütkepohl, H. and J. Wolters, 2003, "The Transmission of German Monetary Policy in the pre-euro Period", Macroeconomic Dynamics, Vol 7, pp. 711-733. Cesar E. Tamayo Stochastic Trends & Economic Fluctuations 14 / 14