THE ECONOMETRICS OF MACROECONOMIC MODELLING

An example

Подпись: Aw ' 0 0' Aw 0 0 Aw Ap t 0.1 0 Ap t-1 0 0.16 Ap Подпись: -0.81 1 Подпись: t-2

Подпись: 0 0.026 Подпись: -0.16 0 image391 Подпись: 0 0 Подпись: -| Aa |- Au + - Api t

As an example, and in the process of illustrating different techniques, we will work out the dynamic properties of the wage-price model of Section 9.2.2. This involves evaluating the stability of the model, and the long-run and dynamic multipliers. Disregarding taxes and short-run effects, the systematic part of the model is on matrix form.

w

Подпись: 1 1 1—1 1 1—1 О i—1 t-ч 0 -0.7 L 1L2 0.7 0 0.3 P

a

u

Подпись: 1РЧ t

Steady-state properties from cointegration The long-run elasticities of the model are, from the cointegration analysis.

w = p + a — 0.1u p = 0.7(w — a)+ 0.3pi,

so the long-run multipliers of the system should be easily obtained by solving for wages and prices. For wages:

Подпись: w w(1 — 0.7) w w 0.7(w — a) + 0.3p + a — 0.1u —0.7a + 0.3pi + a — 0.1u

0.3 0.1

--- a--------- u + pi

0.3 0.3 [118] a — 0.33u + pi.

Then for prices:

p = 0.7(w — a)+ 0.3pi = 0.7(—0.33u + pi) + 0.3pi p = —0.23u + pi.

So the reduced form long-run multipliers of wages and prices with respect to the exogenous variables are

w = a — 0.33u + pi p = —0.23u + pi.

Note that the long-run multipliers of the real wage are given from the wage curve alone

w — p = a — 0. 1 u.

Imposing long-run properties of exogenous variables

• Aa = ga

• Au = 0

• Api = gpi

gives the long-run multipliers for inflation

n = gp = Ap = gpi.

Finally, the steady-state growth path of the nominal system is

gw ga + gpi

gp = gpi.

Dynamic properties from difference equations Now, let us try to see

if this holds for the dynamic system. Intuitively, the same steady state—and therefore the same multipliers—should be obtained if no invalid restrictions are imposed.

For the dynamic analysis of the system below, following Wallis (1977), it will be more convenient to work with the model in lag-polynomial form

Подпись: A(L)A(L)yt = B(L)xt. This is easily achieved with the steps:

1 -0.81

Aw

-0.14 - 0.1L 1 - 0.16L2

Ap

-|

Aa

r

Au

+

-

Api

t

0.082 0 0 -0.015 0 0.026

-0.16 0 0 0.055

L - L - L 0.1L2 0

-0.7L2 L3 0.7L 0 -0.3L

image398

Подпись: “i a u pi Подпись: t

image401 Подпись: 0 0.0165L Подпись: t

or:

image404

and collecting terms:

A (L)

Подпись: 1 a u pi 0.082 + 0.078L -0.016L2 0

-0.015 - 0.0235L 0 0.026 - 0.0095L [119] 1

The model is stable if all the roots of

0.8866 - 1.7166z + 0.703815z2 + 0.309425z3 - 0.1806z4 = 0

are outside the unit circle. Here the polynomial can be factored (approx­imately) as

-0.1806(z + 2.26942781)(z - 1.03041478)(z - 1.19380201)(z - 1.75852774) = 0 so the roots are

{

-2.26942781 I 1.03041478 I 1.19380201.

1.75852774

So all roots of IA (z)| = 0 are outside the unit circle. Also, in this case, the roots are real, so the adjustment from a shock back towards steady state will be monotonic and non-cyclical.

Deriving the long-run multipliers—the hard way Next the long-run multipliers are A-1(1)B(1). Here A(1) is given as:

1 - 0.84 -0.81 + 0.65

-0.14 + 0.04 + 0.0615 1 - 0.16 - 1 +0.215

0.16 -0.16"

0.0385 0.055 ,

while

Подпись: B (1)0.082 + 0.078 -0.016 0

0.015 0.0235 0 0.026 0.0095

0.16 -0.016 0 0.0385 0 0.0165

Подпись: A-1(1)B (1)Подпись: 0.16 0.0385

image409

giving the long-run multipliers

1.0

-0.33

1.0

0

0.23

1.0

or

w

1—1

0

1

о

со

со

1—1 о

a

_p_

=

0 -0.23 1.0

u

which corresponds to the long-run multipliers derived directly from the cointegration analysis.

So the cointegration relationships is therefore the steady-state of the dynamic system; it ties down the long-run solution of the dynamic system, and the comparative static properties—the long-run multipliers. In fact, this is nothing else than Samuelson’s correspondence principle in disguise.

Deriving the long-run multipliers—the easy way To show that coin­tegration is nothing but steady-state with growing variables is just finding the long-run multipliers as in Bardsen (1989), but now for systems. The reduced form of the model is:

Подпись: 0.09 0 Aw + t-1 0 0.146 Aw 0.113 0 Ap 0 0.18 Ap Aw

Подпись: 0.079 -0.004 Подпись: 0 0 image413 Подпись: + n(L) image415 Подпись: 1

Ap

with the cointegration part alone:

Подпись: -0.18 + 0.035L 0.18 - 0.05L2 0.145 -0.018L 0.015' -0.025 + 0.042L 0.025 - 0.06L2 -0.017 -0.0025L 0.018 n(L) image417 Подпись: 1

w

Подпись: -0.145 0.13 0.017 0.035 Подпись: 0.145 -0.018 0.015 0.017 0.0025 0.018 Подпись: w p a u pi t

or when evaluated at the same date, so in steady-state:

The long-run multipliers are therefore simply:

— 1

a

w

-0.145

0.13

0.145

-0.018

0.015

p

0.017

-0.035

-0.017

-0.0025

0.018

u

pi

w

1

1—1

0

CO

CO

1

1—1

a

p

1

о

0 bo

CO

1

1—1

u

pi

as before.

image422

image423

The inverse autoregressive matrix polynomials are of course the product of the inverse of the determinant and the adjoint

 

1

 

1 - 0.84L -0.81 + 0.65L

-0.14 + 0.04L + 0.0615L2 1 - 0.16L2 - 1L + 0.215L3 1

0.89 - 1.72L + 0.7L2 + 0.31L3 - 0.18L4 "1 - 0.16L2 - L + 0.215L3 0.81 - 0.65L 0.14 — 0.04L — 0.0615L2 1 — 0.84L

 

A-1(L)

 

The matrix of distributed lag-polynomials was

 

0.082 + 0.078L -0.016L2 0

-0.015 - 0.0235L 0 0.026 - 0.0095L.

 

B (L)

 

Therefore

 

'^11 (L) ^12 (L) ML)'

M (L) 522 (L) ML)_

1

0.89 - 1.72L + 0.7L2 + 0.31L3 - 0.18L4 "1 - L - 0.16L2 +0.215L3 0.81 - 0.65L X [0.14 - 0.04L - 0.0615L2 1 - 0.84L

" 0.082 + 0.078L -0.016L2 0

X -0.015 - 0.0235L 0 0.026 - 0.0095L

1

0.89 - 1.72L + 0.7L2 + 0.31L3 - 0.18L4

 

D(L)

 

image424

Подпись: (0.02 - 0.02L' +0.006L2) (0.03 - 0.03L +0.008L2)

'(0.07 - 0.01L - 0.08L2 +0.01L3 + 0.02L4) (-0.004 - 0.003L _ +0.01L2 - 0.005L3)

(-0.02L2 + 0.02L3 +0.003L4 - 0.003L5) (-0.002L2 +0.0006L3 +0.001L4)

So to find the dynamic multipliers of wages with respect to productivity S11,i, for period i = 0,1, 2, we have to solve

0.07 - 0.013L - 0.076L2 + 0.005L3 + 0.02L4

= (0.89 - 1.72L + 0.7L2 + 0.31L3 - 0.18L4)(£u,0 + 5пл1 + S11,2L2)

= 0.89£110 + (0.89£цд - 1.72£110)L + (0.89Jn,2 - 1.72J1M + 0.70£u,0)L2 + (-1.725ii,2 + 0.705ii, i + 0.31Sn, o)L3 + (0.705ii,2 + 0.315ii, i - 0.18£ii, o)L4 + (0.31Jii,2 — 0.18^ii, i)L5 — 0.18^ii,2L6

for the J’s by evaluating the polynomials for powers of L:

= 0.079,

11,0 - 0.013

L = 0: L = 1: L = 2:

 

^11,0

£ll, l

&11,2

 

image426

0.119.

 

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[1] RIMINI has been used by the Central Bank of Norway for more than a decade to make forecasts for the Norwegian economy 4—8 quarters ahead as part of the Inflation report of the Bank; see Olsen and Wulfsberg (2001).

[2] Cross (1995, p. 184) notes that an immutable and unchangeable natural rate was not implied by Friedman (1968).

[3] Elmeskov and MacFarland (1993), Scarpetta (1996), and OECD (19976: ch. 1) contain examples.

[4] See Bjerkholt (1998) for an account of the Norwegian modelling tradition.

[5] See Jansen (2002), reply to S0ren Johansen (Johansen 2002).

[6] Theory that attributes the origin of matter and species to a special creation (or act of God), as opposed to the evolutionary theory of Darwin.

[7] Jacobson et al. (2001) use a structural VAR with emphasis on the common trends to analyse the effect of monetary policy under an inflation targeting regime in a small open economy.

[8] As is clear from the discussion above, econometric methodology lacks a consensus, and thus the approach to econometric modelling we are advocating is controversal. Heckman (1992) questions the success, but not the importance, of the probabilistic revolution of Haavelmo. Also, Keuzenkamp and Magnus (1995) offer a critique of the Neyman—Pearson paradigm for hypothesis testing and they claim that econometrics has exerted little influence on the beliefs of economists over the past 50 years; see also Summers (1991). For sceptical accounts of the LSE methodology, see Hansen (1996) and Faust and Whiteman (1995, 1997), to which Hendry (19976) replies.

[9] Naturally, with a very liberal specification strategy, overfitting will result from Gets modelling, but with ‘normal’ requirements of levels of significance, robustness to sample splits, etc., the chance of overfitting is small. Thus the documented performance of Gets modelling now refutes the view that the axiom of correct specification must be invoked in applied econometrics (Leamer 1983). The real problem of empirical modelling may instead be to keep or discover an economically important variable that has yet to manifest itself strongly in the data (see Hendry and Krolzig 2001). Almost by implication, there is little evidence that Gets leads to models that are prone to forecast failure: see Clements and Hendry (2002).

[10] Interestingly, the papers that introduced the limited information methods also introduced the first tests of overidentifying restrictions in econometrics.

[11] Johansen (2002) has pointed out that LIML does not work with cointegrated systems, where relaxing cross equation restrictions (implied by cointegration) changes the properties of the system.

[12] In fact there were two models, a short-term multisector model and the long-term two sector model that we reconstruct using modern terminology in this chapter. The models were formulated in 1966 in two reports by a group of economists who were called upon by the Norwegian government to provide background material for that year’s round of negotiations on wages and agricultural prices. The group (Aukrust, Holte, and Stoltz) pro­duced two reports. The second (dated 20 October 1966, see Aukrust 1977) contained the long-term model that we refer to as the main-course model. Later, there were similar develop­ments in, for example, Sweden (see Edgren et al. 1969) and the Netherlands (see Driehuis and de Wolf 1976).

In later usage the distinction between the short - and long-term models seems to have become blurred, in what is often referred to as the Scandinavian model of inflation. Rpdseth (2000: ch. 7.6) contains an exposition and appraisal of the Scandinavian model in terms of current macroeconomic theory. We acknowledge Aukrust’s clear exposition and distinction in his 1977 paper, and use the name main-course model for the long-run version of his theoretical framework.

[13] On the role of the main-course model in Norwegian economic planning, see Bjerkholt (1998).

[14] In France, the distinction between sheltered and exposed industries became a feature of models of economic planning in the 1960s, and quite independently of the development in Norway. In Courbis (1974), the main-course theory is formulated in detail and illustrated with data from French post-war experience (we are grateful to Odd Aukrust for pointing this out to us).

[15] Note that, due to the log-form, ф = xs/(1 — xs) where xs is the share of non-traded goods in consumption.

[16] See Hendry (1995a: ch. 7.4) on the role of differenced data models in econometrics.

[17] See Aukrust (1977, p. 130).

[18] The main current of theoretical work is definitively guided by the search for ‘microfounda­tions for macro relationships’ and imposes an isomorphism between micro and macro. An interesting alternative approach is represented by Ferri (2000) who derives the Phillips curve as a system property.

[19] Alternatively, given H2mc, Awt represents the average wage growth of the two sectors.

[20] The rate of unemployment enters linearly in many US studies; see, for example, Fuhrer (1995). For most other datasets, however, a concave transform improves the fit and the stability of the relationship; see, for example, Nickell (1987) and Johansen (1995a).

[21] To affect - uphl1, policy needs to incur a higher or lower permanent rate of currency depreciation.

[22] Hence, the first term in (4.9) reflects normal cost pricing in the sheltered sector. Also, as a simplification, we have imposed identical productivity growth in the two sectors,

Aae, t = Aaa, t = Aat.

[23] This section draws on Ericsson et al. (1997).

[24] Super exogeneity is defined as the joint occurrence of weak exogeneity of the explanatory variables with respect to the parameters of interest and invariance of the parameters in the conditional model with respect to changes in the marginal models for the explanatory variables, see Engle et al. (1983).

[25] Below, and in the following, square brackets, [..], contain p-values whereas standard errors are stated in parentheses, (..).

[26] The dummy variable IPt is designed to capture the effects of the wage-freeze in 1979 and the wage-laws of 1988 and 1989. Similar dummies for incomes policy appear with significant

[27] The numbers refer to the ‘total’ rate of unemployment, that is, including persons on active labour market programmes.

[28] Residual standard deviations and model diagnostics are reported at the end of the table. Superscript v indicates that we report vector versions of the single equation mis-speciflcation tests encountered above, see equation (4.42). The overidentiflcation x2 is the test of the model in Table 4.2 against its unrestricted reduced form, see Anderson and Rubin (1949, 1950), Koopmans et al. (1950), and Sargan (1988, pp. 125 ff.).

Note: The sample is 1964 to 1998, T = 35 observations &Aw = 0.014586 aAtu = 0.134979 &Ap = 0.0116689 FAR(i-2)(18, 59) = 1.0260[0.4464]

N2ormality(6) = 3.9186[0.6877] x2vendentfication(36)= 65.533[0.002]

the rate of unemployment will take a long time before it eventually returns to the natural rate, thus confirming Figure 4.4.

Figure 4.5 offers visual inspection of some of the dynamic properties of the model. The first four graphs show the actual values of Apt, tut, Awct, and the wage share wct — qt — at together with the results from dynamic simulation. As could be expected, the fits for the two growth rates are quite acceptable. However, the ‘near unit root’ property of the system manifests itself in the graphs for the level of the unemployment rate and for the wage share. In both cases there are several consecutive years of under - or overprediction. The last two displays contain the cumulated dynamic multipliers of tu and the wage share resulting from a 0.01 point increase in the unemployment rate. As one might expect from the characteristic roots, the stability property is hard to gauge from the two responses. For practical purposes, it is as if the level of unemployment and the wage share ‘never’ return to their initial values. Thus, in the model in Table 4.2, the equilibrium correction is extremely weak.

[29] See Wallis et al. (1984, p. 134).

[30] Nevertheless, the ICM acronym may be confusing—in particular if it is taken to imply that the alternative model (the Phillips curve) contains perfect competition.

[31] R0dseth (2000: ch. 8.5) contains a model with a richer representation of the demand side than in the model by Layard et al. (1991). R0dseth shows that the long-run equilibrium must satisfy both a zero private saving condition and the balanced current account condition.

[32] Recall that we expressed the Nash-product as

7?____ VW, U,Zv)_____ _ /і _ 7П 1/A

° v(Wq/Pq, U,Zv ) - V0(Wq/Pq, U) _(± U> (1 - Wq/A) >

in (5.3).

[33] See their equation (4), which uses the lagged real wage, which cointegrates with current real wage, on the right-hand side.

[34] The statistics reported in the table are explained in Section 4.6, Table 4.2, and in connection with equation (4.43).

[35] This result is the opposite of Rpdseth and Holden (1990, p. 253), who found that deviation from the main course is corrected by Amct defined as Aat + Aqt. However, that result is influenced by invalid conditioning, since their equation for Amct has not only ecmt—i, but also Awct on the right-hand side. Applying their procedure to our data gives their results: for the sample period 1966—98, ecmt—i obtains a ‘f-value’ of 2.94 and a (positive) coefficient of 0.71. However, when Awct is dropped from the right-hand side of the equation (thus providing the relevant framework for testing) the ‘f-value’ of ecmt—i for Aat falls to 0.85.

[36] Note that these estimates are conditioned by the restrictions on the loadings matrix explained in the text and that the the signs of the coefficients are reversed in the graphs.

[37] Norwegian economists know such models as ‘Haavelmo’s conflict model of inflation’, see Qvigstad (1975).

[38] Haavelmo formulated his model, perhaps less deliberately, in terms of two separate target real wage rates for workers and firms (corresponding to wb and wf of Chapter 5), but the implications for inflation are the same as in Rowthorn’s model.

[39] See Kolsrud and Nymoen (1998) for an explicit parameterisation with nominal variables with long-run homogeneity imposed.

[40] Layard et al. (1994, p. 18), authors’ italics.

[41] Of course, if there is a long-run effect of competitiveness on prices, that is (5.6) is extended by a competitiveness term, ш = 0 is not sufficient to produce an unstable solution.

[42] See, for example, Layard et al. (1991, p. 391).

[43] The roots of the system (where ut is exogenous) are rx = 1 — 9q and Г2 = 1.

[44] Note that an identical line of reasoning starts from setting 9q = 0 and leads to a price Phillips curve NAIRU. This seems to give rise to an issue about logical (and empirical) indeterminacy of the NAIRU, but influential papers like Gordon (1997) are not concerned with this, reporting instead different NAIRU estimates for different operational measures of inflation.

[45] From Dreze and Bean (1990, table 1.4), and the country papers in Dreze and Bean (1990) we extract that the equations for Austria, Britain, and (at least for practical purposes) Germany are ‘true’ product real-wage equations. The equation for France is of the Phillips - curve type. For the other countries we have, using our own notation: Belgium and the Netherlands: consumer real-wage equations, that is, pwp = 1, pwq = 0, and ш = 1. Denmark: ш = 1, ф-шр = 0.24, pwq = 0.76. Italy: ш = 0, pwp = 0.2(1 — ф), pwq = 0.8(1 — ф). United States: ш = 0.45(1 — ф), pwq = 1, pwp = 0. Spain: ш = 0.85• 0.15, 0w = 1, pwp = ш, Ф-wq = 1 — ш (the equation for Spain is static).

[46] See Rowlatt (1992: ch. 3.6).

[47] The analysis follows Holden and Nymoen (2002).

[48] The full quotation is given in Section 4.6.

[49] Note that in the Norwegian Phillips curve of Section 4.6 and in Section 6.9.2, the log of the total unemployment rate was used. In the cross-country results reported here we chose to use open unemployment for all countries. However, as documented in Nymoen and Rpdseth (2003), the choice has little influence on the estimation results.

[50] The appearance of this variable has to do with the use of the open rate of unemployment, rather than the total rate.

[51] The other elasticities in (6.53) are also non-negative.

[52] This equation is similar to (4.9) in the Phillips curve chapter. The only difference is that we now let import prices represent imported inflation.

[53] The inflation rate depends on Awct, a feature which is consistent with the result about an endogenous real-wage wedge in the cointegration analysis of Chapter 5, Section 5.5: pt — qt was found to be endogenous, while the product price (qt) was weakly exogenous.

[54] This chapter draws on Bardsen et al. (2002b, 2004).

[55] The overlapping wage contract model of sticky prices is also attributed to Phelps (1978).

[56] That is, subject to the transversality condition limn^^(bpi)n+1 Apt+n+1 = 0.

[57] The full set of coefficient values are: bxi = 0, = 0.25, bbpl = 0.75, bx2 = 0.7.

[58] See Bardsen et al. (2002b) for a more detailed discussion.

[59] We used the default GMM implementation in Eviews 4.

[60] The rule of thumb is a value bigger than 10 in the case of one endogenous regressor.

[61] David F. Hendry suggested this test procedure to us. Bj0rn E. Naug pointed out to us that a similar procedure is suggested in Hendry and Neale (1988).

[62] Although we use the same set of instruments as Batini et al. (2000), we are unable to replicate their table 7b, column (b). Inflation is the first difference of log of the gross value added deflator. The gap variable is formed using the Hodrick—Prescott (HP) trend; see Batini et al. (2000) (footnote to tables 7a and 7b) for more details.

[63] Inflation Apt in equation (7.17) is for the gross value added price deflator, while the price variable in the study by Bardsen et al. (1998) is the retail price index pct. However, if the long-run properties giving rise to the ecms are correct, the choice of price index should not matter. We therefore construct the two ecms in terms of the GDP deflator, pt, used by Batini et al. (2000).

[64] The conclusion is unaltered when the two instruments are defined in terms of pct, as in the original specification of Bardsen et al. (1998).

[65] Inflation is measured by the official consumer price index (CPI).

[66] In other studies, such direct effects from money aggregates (or measures derived from them) are rejected, cf. for example, de Grauwe and Polan (2001) who argue that the seemingly strong link between inflation and the growth rate of money is almost wholly due to the presence of high (or hyper-) inflation countries in the sample. Similarly, Estrella and Mishkin (1997) reject the idea that broad money is useful as an information variable and provide a good signal of the stance of monetary policy, based on their analysis of United States and German data.

[67] See Clements and Hendry (1998) and Chapter 11.

[68] Concepts and definitions used by Norges Bank to compile Monetary Statistics are now in line with the guidelines in the Monetary and Financial Statistics Manual (MFSM) of the International Monetary Fund (IMF).

[69] The aggregated data underlying AWM are constructed by using a set of fixed purchasing power parity (PPP) exchange rates between the national currencies, calculated for the year 1995, to convert all series to a common currency (i. e. Euro). An alternative aggregation method has been suggested by Beyer et al. (2001) (see also Beyer et al. (2000)). They argue that aggregation across individual countries is problematic because of past exchange rate changes. Hence, a more appropriate method, which aggregates exactly when exchange rates are fixed, consists in aggregating weighted within-country growth rates to obtain euro­zone growth rates and cumulating this euro-zone growth rate to obtain aggregated levels. The aggregate of the implicit deflator price index coincides with the implicit deflator obtained from the aggregated nominal and real data.

[70] Coenen and Wieland adopt a system approach, namely an indirect inference method due to Smith (1993), which amounts to fitting a constrained VAR in inflation, the output gap and real wages, using the Kalman filter to estimate the structural parameters such that the correlation structure matches those of an unconstrained VAR in inflation and the output gap.

[71] See Hodrick and Prescott (1997).

[72] Trecroci and Vega estimate the P*-model within a small VAR, which previously has been analysed in Coenen and Vega (2001).

[73] The first three are significant in all estimated equations reported below, the last two which originate in the AWM wage equation are always insignificant.

[74] The series is extended with data from an internal ECB data series for M3 (M. U2.M3B0.ST. SA) which matches the data of Gerlach and Svensson (2003) with two exceptions, as is seen from Figure 8.8.

[75] We use A = 1600 to smooth the output series y and A = 400 to smooth the velocity v.

[76] We have considered two alternative reference paths for inflation: it is either trend inflation from a smoothed HP filter, or as the same series with the reference path for the price (target) variable of Gerlach and Svensson (2003) substituted in for the period 1985(1)— 2000(2). It is seen that the alternative reference path series share a common pattern. Here we report results based on the first alternative.

[77] Rudd and Whelan (2004) show that including Apt—i among the instruments leads to an upward bias in the coefficient of the forward variable; see also Roberts (2001). We have, however, maintained the use of the Gall et al. (2001) instruments simply to get as close as possible to the estimation procedure adopted by the ‘proprietors’ of the NPCM in the same way as we have tried to do in the cases of AWM price block and the P*-model earlier.

[78] Our estimation method thus differs from those in Chapter 7, where we estimate the hybrid model using generalised method of moments (GMM) as well as by two-stage least squares. Note that we in Chapter 7, like Gall et al. (2001), use the GDP deflator while in this section the inflation variable is the consumption deflator.

[79] First-order autocorrelation may also have other causes, as pointed out Chapter 7.

[80] For an introduction to the encompassing principle, see Mizon and Richard (1986) and Hendry and Richard (1989).

[81] It should be noted that the encompassing tests FEncGum, reported in Tables 8.7 and 8.8, are based on two-stage least squares estimation of the NPCM. This gives estimates of the inflation equation that are close to, but not identical to, those in equation (8.22), since full information maximum likelihood (FIML) takes account of the covariance structure of the system. In order to form the minimal nesting model it was necessary to estimate the NPCM on a single equation form to make it comparable with the other (single equation) models.

[82] Strictly speaking, the generic GUM is the union of all information sets we have used to create the general models in Sections 8.6.1—8.6.4. In the minimal nesting (parsimonious) GUM, we have left out all variables that are not appearing in any of the five final equations and it is more precise to call this a pGUM.

[83] Again, the forecast encompassing tests are based on two-stage least squares estimates of the NPCM.

[84] A more realistic approach would have been to let the estimates of the equilibrium values be derived from some backward-looking filter. Such a procedure would better capture the relevant information available to the forecaster when forecasts are made.

[85] This point is, however, not relevant to the P*-model in its original tapping (see Hallman et al. 1991), where weight is put on the quantity equation and the stability of the money demand function. Fagan and Henry (1998) suggest that money demand may be more stable at the aggregated Euro-area level than at the national levels.

[86] The final inflation equation also includes short-run effects of changes in the length of the working day (Aht) and seasonal dummies.

[87] It should be noted that the encompassing tests FEncGum, reported in Table 8.18, are based on two-stage least squares estimation of the NPCM. In order to form the minimal nesting model it was necessary to estimate NPCM on a single equation form to make it comparable to the other (single equation) models.

[88] Strictly speaking, the generic GUM is the union of all information sets we have used to create the general models in Sections 8.7.1—8.7.3. In the minimal nesting (parsimonious) GUM we have left out all variables that are not appearing in any of the five final equations and it is more precise to call this a pGUM.

[89] From the previous sections we have seen that many of the models automatically provide forecasts of annual inflation since A4pt is the left-hand side variable. In all models of this type we have included Azpt—i unrestrictedly as a right-hand side variable. If the coefficient of Азр(_і is close to one, the annual representation is a simple isomorphic transformation of a similar quarterly model. The NPCM is only estimated with quarterly inflation, Apt, as left-hand side variable. Thus, for the purpose of model comparison we have re-estimated all models with Apt as left-hand side variable.

[90] The caveat mentioned in Section 8.6.6 of the P*-model being greatly helped by the use of two-sided HP-filters is also relevant for the case of Norway.

[91] Recall the caveat in Section 8.6.6—that the P*-model is unduly helped by the use of two-sided filters—which further strengthens the case for the AWM.

[92] We model the mainland economy only, although the oil sector accounts for close to 20 per cent of total GDP. The oil activities, including the huge oil investments, are driven by factors that are exogenous to the mainland economy, which we have chosen to focus on.

[93] In other words, a formal inflation target was introduced at the end of the last quarter included in the sample.

[94] In Chapter 10 we analyse the performance of different monetary reaction functions.

[95] A precursor to the model can be found in Bardsen et al. (2003). Other comparable econometric studies are Sgherri and Wallis (1999), Jacobson et al. (2001), and Haldane and Salmon (1995)—albeit with different approaches and focus.

[96] The marginal models reported below are estimated with OLS.

[97] In effect we model the exchange rate, treating foreign prices as being determined by factors that are a priori unrelated to domestic conditions.

[98] The idea to first let the marginal models include non-linear terms in order to obtain stability and second to use them as a convenient alternative against which to test invariance in the conditional model, was first proposed by Jansen and Terasvirta (1996).

[99] There is no marginal model for the impact of import prices Apit. Instead, we have assumed full and immediate pass-through of the exchange rate, imposing Apit = Avt + Apwt on the model. We therefore use the intervention variables of Avt to test for invariance of the parameters of Apit.

[100] This is a convenient model simplification, implicitly treating the money market rate as if there is an instant pass-through of a change in the signal rate of the central bank.

[101] This chapter draws on Akram et al. (2003).

[102] Walsh’s results are based on simulations from a calibrated stylised New-Keynesian model. The forecasting properties of the New Keynesian Phillips curve are compared with those of alternative inflation models (on data for Norway and for the Euro area) in Chapter 8.

[103] The model of Chapter 9 is therefore supplemented with a technical equation linking headline inflation (Apt) and underlying inflation (Aput), which is the inflation measure entering the reaction functions of this chapter. Aput measures inflation net of changes in energy prices and indirect taxes.

[104] These are ‘real-time’ variables in the sense that reliable current-quarter information is either available or arrives with only a short time lag; see Orphanides (2001).

[105] A recent example is Levin et al. (2003). In their study of the United States economy they consider (optimised) forecast-based interest rate rules of the type

RSt = UrRS_i + (1 - Ur)(RR* + fit+в) + Up(fit - Пф) + Uyyf+K,

where yf+K is a model-based forecast of the output level к periods ahead and all other symbols are as defined in the main text.

For any given values of (RR*, n*) each rule is fully described by the triplet (ur, Up, Uy), and Levin et al. (2003) derive the parameters of such interest rate rules for five different models under the assumption that the Central Bank’s preference function is given by

£(X) = У[nt] + XV[ygap], subject to V[ARSt] < v2ARs, A € (0,1/3, 1, 3),

This loss function is then minimised subject to an upper bound on the volatility of the interest rate, o'2 .

’ 2rs

[106] For the real exchange rate vr the trigger value of the target is 0. Hence (vr — vr*) is equivalent to deviations from purchasing power parity (PPP), (v + pw — p), cf. Section 9.3.1.

[107] In the baseline simulation the model residuals have been calibrated such that the actual values of the data are reproduced exactly when we simulate the model with historical values for the short-run interest rate, RSt. For each of counterfactual simulations with the different

[108] From the definition of b(h—2) 'n (11-25) it follows that b(h—3) = Ф(к—2) — Inserting this in the recursive formula for ^(h—3) and rearranging terms yields a^(h—2) = (h — 1) — (1 — a)S(h—2). Finally, when we add S(h—1) on both sides of this equality and apply the recursive formula for S(h—1) in (11.25), the expression simplifies to (h — 1) + 1 = h.

[109] The underprediction of consumption expenditures in Norway during the mid-1980s, which marred Norwegian forecasters for several consecutive forecasting rounds at that time, is a relevant example; see Brodin and Nymoen (1989, 1992). Eitrheim et al. (2002b) give a detailed analysis of the breakdown and reconstruction of the Norwegian con­sumption function that took place in the wake of these forecast failures, and show that what happened can be explained in the light of forecasting theory, see Section 2.4.2.

[110] See Section 1.4, in this application we have used Version 2.9 of the model. A large share of the 205 endogenous variables are accounting identities or technical relationships creating links between variables; see Eitrheim and Nymoen (1991) for a brief documentation of an earlier version of the model.

[111] Abstracting from the problem that the information sets differ across the models consid­ered, and apart from the fact that we use the empirical RMSFE (rather than the theoretical), ranking of the models according to RMSFE is the same as ranking by the squared bias. For a more comprehensive analysis of the use of RMSFEs for model comparisons and the potential pitfalls involved, see for example, Ericsson (1992), Clements and Hendry (1993).

[112] Compared to the algebraic sections of Chapter 6, we omit productivity. Naturally, it is included as a non-modelled explanatory variable in the empirical models.

bThe sample is 1967(1)-1994(4), 112 observations.

The PCM When estimating a PCM, we start out from the same informa­tion set as for the ICM, but with more lags in the dynamics, to make sure we end up with a data-congruent specification. This is to secure that the forecast comparison below is not harmed by econometric mis-specification. It is not implied that the resulting model, given in (11.55), would be seen as the preferred choice if one started out (possibly from another information set) with the aim of finding the best PCM, also in terms of economic interpretation.7 As the diagnostic tests in Table 11.4 show, the model encompasses its reduced form and shows no sign of mis-specification. The estimated standard errors, however, are for both equations higher than the corresponding ones found in the ICM.

7 Dynamic price homogeneity in the wage Phillips curve cannot be rejected statistically, and is therefore imposed.

[114] - bpia

[115] See, for example, Chiang (1984, p. 506).

[116]

, Ept+i - » 7

a2 a2 - bx

[117]

-0.14

[118] - 0.84L -0.81+0.65L

0.14 + 0.04L + 0.0615L2 1 - 0.16L2 - 1L + 0.215L3 ’

with determinant:

|A(L)| = 0.8866 - 1.7166L + 0.703815L2 + 0.309425L3 - 0.1806L4.

B (L)

Checking stability For the system to be stable, the autoregressive part needs to have all roots outside the unit circle.

The autoregressive polynomial is

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THE ECONOMETRICS OF MACROECONOMIC MODELLING

Inflation equations derived from the P*-model

The P*-model is presented in Section 8.5.4. The basic variables of the model are calculated in much the same way for Norway as for the Euro area in the previous …

Forecast comparisons

Both models condition upon the rate of unemployment ut, average labour productivity at, import prices pit, and GDP mainland output yt. In order to investigate the dynamic forecasting properties we …

The NPCM in Norway

Consider the NPCM (with forward term only) estimated on quarterly Norwegian data[65]: Apt = 1.06 Apt+1 + 0.01 wst + 0.04 Apit + dummies (7.21) (0.11) (0.02) (0.02) x2(10) = …

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