The Nominal-to-Real Transformation

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1 The Nominal-to-Real Transformation Hans Christian Kongsted PhD course at Sandbjerg, May 2 Based on: Testing the nominal-to-real transformation, J. Econometrics (25) Nominal-to-real transformation: Reduce the order of integration of an I(2) vector time series. Retain all cointegrating relations among variables. Examples of real-transformed systems: Real money and in ation, e.g. KPSW: m t p t = p t + z t + u t The markup and the rate of inflation, e.g. Banerjee et. al.: p t c t = 2 p t + 2 z t + u 2t N-t-R may fail in two directions with di erent implications for the real transformed system: Analytical results. Tests of the N-t-R transformation: Sequential test in two-step procedure Joint test in MLE procedure Transformed model I(2) test: Misspeci cation-type approach in real transformed system. Simulation experiment Empirical evidence

Examples of real-transformed systems: Real money and in ation, e.g. KPSW: m t p t = p t + z t + u t The markup and the rate of inflation, e.g. Banerjee et. al.

2 I(2) Cointegration in the VAR 2 X t = X t X t + " t ; t = ; : : : ; T; " t i:i:d:n(; ): Reduced rank conditions: {z} = and?? = : {z } rank r<p rank s<p r Define =? and 2 =??. Define =? and 2 =??. No higher order of integration: 2 2 full rank. MA representation: X t = C 2 tx s= sx tx " i + C " i + stat. part + initial cond. i= i= C 2 = 2 ( 2 2 ) 2 : rank p r s. X t : r linear combinations, CI(2,) and polynomially cointegrating: X t 2 2X t is I(). X t : s linear combinations, CI(2,). 2

MA representation: X t = C 2 tx s= sx tx " i + C " i + stat. part + initial cond. i= i= C 2 = 2 ( 2 2 ) 2 : rank p r s.

3 N-t-R transformation Hypothesis: Known matrix b defines I(2) loadings: sp(b) = sp( 2 ): Equivalently: All CI(2,) cointegrating relations are subject to p r s linear restrictions: Propose transformation: b (; ) = : ~X t = B X t v X t = Zt with B = b? and jv bj 6=. Eliminates I(2) trend, retains cointegrating combinations if and only if b (; ) =. Money demand analysis: X t = (m t ; p t ; y t ; R t ) ; Xt ~ = (m t p t ; y t ; R t ; p t ) : b = B A ; B = B A ; v = B A : U t Markup-inflation analysis: X t = (p t ; c t ; p mt ) ; Xt ~ = (p t c t ; p t p mt ; p t ) : b A ; B A ; v A : 3

Eliminates I(2) trend, retains cointegrating combinations if and only if b (; ) =.

4 Example Adopted measurements of three nominal variables: The money stock, m t, nominal income, y n t, and the price level, p t. Assume that X t = (m t ; y n t ; p t ) is I(2) with r = s =. There is one I(2) common trend in the nominal system. Further assumptions maintained by all cases: The price level, p t, is I(2). There exist real numbers and de ning a generalized velocity relationship in which m t yt n + p t cointegrate to stationarity. Theory: m t, y n t and p t are equally a ected a single nominal trend, b = (; ; ). Real transformed variables are ~X t = (m t p t ; y n t p t ; p t ) Choices of B and v: B A ; v A : 4

Further assumptions maintained by all cases: The price level, p t, is I(2).

5 Example: Three cases Table : Three special cases of the simulation model. Case Case Case 2 = ; = 6= ; = = ; 6= 2 ( + 2 ) 2 2 Case : The adopted measurements satisfy the theoretical presumption. Real money, m t p t, and real income, yt n p t, are integrated of order (at most) one. The real magnitudes, in turn, cointegrate to stationarity with = in the standard measure of inverse velocity, m t yt n (possibly with in ation). Case : A case in which p t is the appropriate de ator in nominal income whereas the adopted money stock measure requires disproportional price adjustment. Real income, yt n p t, is I() whereas m t and p t cointegrate with a parameter 6=. Real money, m t p t, and standard inverse velocity, m t yt n, are I(2) processes in this case. Case 2: A case in which nominal income corresponds to the adopted measure of money in the sense that the standard inverse velocity measure, m t yt n, cointegrates to I(). The price measure adopted, p t, is not proportional either in money or income. Real money as well as real income remain I(2) processes and X ~ t is I(2). Check the condition b (; ) = for each case! 5

The real magnitudes, in turn, cointegrate to stationarity with = in the standard measure of inverse velocity, m t yt n (possibly with in ation).

6 Properties of the real-transformed system If b =, then transformed VECM exists in terms of ~X t : Picks up right levels combinations of variables. Full set of rst-di erences by jv bj 6=. Multiply model by (B; v), use X t = cz t + au t with c = V (B V ), V = v?, a = b(v b). 2 Z t U t B = X t v X t B (cz t + au t ) v (cz t + au t ) B + " t v " t : Rearrange terms and use that by b = we have that = B' and thus X t = ' Z t : Zt B = ' B a Zt U t v ' v a U t I B c Zt + B v + " t c U t v " t 6

Multiply model by (B; v), use X t = cz t + au t with c = V (B V ), V = v?, a = b(v b).

7 Properties of the real-transformed system (2) Transformed VECM: X ~ B t = ' B a v ' v a Zt U t + ~ ~ X t +~" t : Apply the relation (Paruolo and Rahbek, 999): = +?? + Matrix of levels coe icients, ~ : ~ = (B; v) [' ;??b(v b) b(v b) ] ' = (B; v) (; )??b(v b) b(v b) : In general: Rank determined by b, r rank( ~ ) r + s < p. Special case b = : rank( ~ ) = r. Standard form of polynomially cointegrating relation obtains as combination of transformed variables, Z t and U t, when v is chosen as b. 7

?b(v b) b(v b) ] ' = (B; v) (; )??b(v b) b(v b) : In general: Rank determined by b, r rank( ~ ) r + s < p.

8 Testing: Sequential test in nominal system Test of sp(b) = sp( 2 ) ts into two-step estimation procedure: Repeated use of I() reduced rank regression (Kongsted, 23). First step: Estimate unrestrictedly and subject to restriction on. LR test Q b asympt. distributed 2 ((p r s)r) subject to b =. Second step conditioned on first-step restricted estimate of : Estimate unrestrictedly and imposing b = as restriction on. Given first-step estimates LR test Q b2 is asympt. distributed 2 ((p r s)s) subject to b (; ) =. Overall test: Rejection region is union of individual test rejection regions with size =2. True asymptotic size of the overall test is between =2 and. Sequential procedure consistent under the alternative b (; ) 6=. 8

Second step conditioned on first-step restricted estimate of : Estimate unrestrictedly and imposing b = as restriction on. Given first-step estimates LR test Q b2 is asympt.

9 Testing (2): MLE-based test Overall test of b (; ) = available as b = in MLE estimation procedure. Combine test of b = and b =, sequentially, to address the source of any failure of the N-t-R transformation. Transformed model I(2) test: Second step of two-step estimation procedure applied to ~ X t. ~ X t is I() if and only if b (; ) =. Address ~s = rank(~? ~ ~? ) with null that ~s = p ~r against full rank alternative, ~s = p ~r. Reverse relationship: Invalid transformation under the null. Valid under full rank alternative. Knowledge of r and s allows construction of sequential test, but provides only lower and upper bounds on ~r. If ~r estimated in transformed model: Actual size exceeds nominal level in the limit if I(2) trend is present. 9

Transformed model I(2) test: Second step of two-step estimation procedure applied to ~ X t. ~ X t is I() if and only if b (; ) =. Address ~s = rank(~? ~ ~?

10 Simulation experiment The data generation process (DGP) for the experiment is de ned by the equations X t X 2t = ( )X 2t +(( ) + )X 3t + " t " 2t X 2t = X 3t + " 2t 2 X 3t = " 3t (" t ; " 2t ; " 3t ) i:i:d:n(; I) with = + 2 ( + 2 ) 6=. Encompasses the money demand example with X t = m t, X 2t = y n t, and X 3t = p t. In this interpretation, () is a generalized velocity equation, () is the equation for nominal income and () is the price equation. Can be written as a VECM with r = and s =. X t is I(2) and the CI(2,) cointegrating vectors are = (; ; ) and = (; ; ( + 2 )). The vector of I(2) loadings is 2 = (; ; ) and the polynomially cointegrating relationship is S t = X t X 2t (X t +X 2t +X 3t ) in the standard representation.

In this interpretation, () is a generalized velocity equation, () is the equation for nominal income and () is the price equation. Can be written as a VECM with r = and s =.

11 Table 2 Simulation results. = :5.. replications Rejection frequency Rank determination T Q b (Q b ;Q b2 ) seq r = r =2 r =3 Case Case a a a a a a a a a a a a a a a 5.. a a a a a a a a a a a a 5.. a a a a a a Case Note: Q b is the Johansen (995) likelihood ratio test of b =. (Q b ;Q b2 ) seq is the sequential test of b (; ) =, see Kongsted (23). Rank Determination uses the procedure described by Johansen (992b) based on the trace test of Johansen (988). a No VEC representation in terms of X t for Case.

52.5.998 5.86.93.923.68.9 5.6.69.94.53.6 5.53..927.64.9.995 5.77.94.98.72.9 5.62.98.929.63.8 5.53.335.788.83.28.99 5.77.3.93.74.2 5.55.22.869..9 5.52.675.465.479.55.98 5.8.23.878.4.8 5.57.489.664.294.

12 Table 3 Main characteristics of the data Country Sample Variable denitions Source m t p t y t R t U.S M Deator of net Net national Commercial Stock and Watson (993) national product product paper rate U.K Broad Deator of net Net national Short-term Ericsson et al. (998) money national income income interest rate Italy M2 Cost-of-living Real GDP Long-term Muscatelli and Spinelli (2) index government bond yield Deposit rate Note: The Stock and Watson (993) data set was obtained from Mark Watsons homepage. David Hendry kindly supplied the data used in Ericsson et al. (998). The appendix to Muscatelli and Spinelli (2) lists their data. Table 4 Tests of the nominal-to-real transformation Country v ;v 2 Constant and trend Constant Q b (v ) Q b2 (v 2 ) Q b (v ) Q b2 (v 2 ) U.S.,2.68 [.2] 2.93 [.23].67 [.2].34 [.5] U.K.,2.2 [.73] 2.8 [.25] 3. [.8].6 [.45] Italy,3.4 [.7] 36.2 [.] 3.52 [.6] [.] Note: v and v 2 denote the degrees of freedom of the tests. Values in brackets are p-values.

(998) money national income income interest rate Italy 86 996 M2 Cost-of-living Real GDP Long-term Muscatelli and Spinelli (2) index government bond yield Deposit rate Note: The Stock and Watson

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