Recitation 7: Empirics and Theory on Monetary Policy Design

Transcription

1 4.46: Advaned Maroeonomis I Suman S. Basu, MIT Reitation 7: Empiris and Theor on Monetar Poli Design Over the last ouple of weeks in letures, we have onsidered optimal monetar poli in the baseline model and in models with extensions. Professor Gali provided an empirial overview of monetar poli rules in di erent periods, for example the US experiene pre- and post-volker. In the ontext of ost-push shoks, we disussed monetar poli with and without ommitment. This reitation handout overs two papers b Clarida, Gali and Gertler QJE 2000 and JEL 999), and we examine the topis mentioned above in more detail.. Monetar Poli Rules and Maroeonomi Stabilit: Evidene and Some Theor CGG, QJE 2000) Clarida, Gali and Gertler 2000) begin b using GMM in order to estimate the oe ients on monetar poli rules for the US Federal Reserve during pre- and post-volker periods. Then a theoretial model is used in order to predit two real-world onsequenes of the hange in the monetar poli rule: i) sunspot utuations and ii) volatilit with respet to fundamental shoks. The authors propose that this aounts for the di erential performane of in ation and output during the pre- and post-volker periods.. GMM Estimation of a Forward-Looking Rule The authors postulate the following linear equation for the monetar poli rule: r t = r + E[ t;k j t ] ) + E[x t;q j t ] ) where t;k denotes the average value of in ation between periods t and k, and x t;q denotes the equivalent term for the deviation of output from target between periods t and q. Suh a rule is optimal for a entral bank with a loss funtion that is quadrati in the deviations of in ation and output from their respetive targets. Empiriall, suh rules have been fairl suessful in desribing monetar poli as pratised b the US Federal Reserve. r is the target nominal interest rate. To make the model more onsistent with atual US monetar poli, it is assumed that the atual nominal interest rate follows a partial adjustment proess: r t = L)r t + )r t 2) Eah period the Federal Reserve adjusts the nominal rate to eliminate ) of the gap between its urrent target level and some linear ombination of its past values. Combining ) and 2): r t = ) frr ) + t;k + x t;q g + L)r t + " t 3) where " t ) f t;k E[ t;k j t ]) + x t;q E[x t;q j t ])g and rr = r is the long run equilibrium real rate. Notie that the error term is a linear ombination of foreast errors and is therefore orthogonal to an variable in the information set t. Let z t denote a vetor of instruments known when r t is set. Equation 3) then implies the following set of orthogonalit restritions: E fr t ) frr ) + t;k + x t;q g L)r t ) z t g = 0 4) This is the restrition used in the GMM estimation. In the absene of further assumptions we an estimate rr ), but not rr or separatel. Imposing the assumption that the observed sample average value of the real interest rate is rr. This enables us to estimate.

2 .2 Evidene on Poli Reation Funtions Pre- and Post-Volker The historial evidene on eonomi performane is summarized as follows: Baseline estimates for the ke parameters of the monetar poli rule ; ; ; ) are presented for the ase k = q =. CBO estimates of the output gap as used for x t. The evidene points to substantial di erenes in the poli reation funtion between periods. Most importantl, the oe ient for is far below during the pre-volker period and far above during the post-volker period. The estimate for is onl marginall signi ant for the post-volker period. The estimate for the smoothing parameter is high in both periods, suggesting onsiderable interest rate inertia: onl between 0 and 30 perent of a hange in the interest rate target is re eted in the nominal interest rate in the quarter of the hange. The paper onduts further robustness analsis..3 Model of US Eonom The authors use a baseline model that is a version of those found in King and Wolman 996), Woodford 996, 998) and Yun 996), among others. The equilibrium onditions are log-linearized around a zero in ation stead state to ield: t = E[ t+ j t ] + t z t ) 5) t = E[ t+ j t ] r t E[ t+ j t ]) + g t 6) rt = E[ t+ j t ] + x t 7) r t = r t + )r t 8) t z t is the deviation of output from the natural value. We restrit ourselves to k = q =. 2

3 .4 Sunspot Flutuations If the oe ient is far below unit, the equilibrium will be indeterminate. In this sense, the poli feedbak rule itself ma be a soure of maroeonomi instabilit. Under indetermina there ma be sunspot utuations, i.e. maroeonomi utuations that our beause the are expeted to our and not due to hanges in fundamentals. The intuition is as follows. If is less than, an inrease in expeted in ation indues a rise in the nominal interest rate from 7), but not of a su ient magnitude to prevent the real interest rate from falling. The fall in the real interest rate stimulates the eonom and generates higher in ation, from equation 5). Therefore the expetations of higher in ation are self-ful lling. What do these sunspot utuations look like? The authors draw the "sunspot shoks" from a standard Normal distribution, as detailed in their 997 working paper. The impulse response to a sunspot shok are as follows: 3

4 The simulated time series are presented: These are not too dissimilar to the atual behaviour of the US eonom. 4

5 .5 Near-Indetermina and Fundamental Shoks In some of the spei ations in the paper, the standard errors for the estimate of are too large to rule out that its true value is unit. In this ase, it is possible that the eonom is just outside the region of indetermina. However, this does not mean that there will be no di erene in the performane of the eonom between the pre- and post-volker regimes. Even though sunspot utuations are no longer feasible in the eonom, in an eonom bu etted b shoks in ation will be more unstable in the regime where the estimate of is lower, i.e. in the pre-volker regime. The lial response of the eonom to fundamental shoks is quite sensitive to. The inrease in from to 2 auses the volatilit of both output and in ation to fall b more than a half, for both suppl and demand shoks. If is low, the entral bank omes lose to full aommodating the in ationar shok, and the absene of a stabilizing movement from the real rate of interest means that the eonom utuates more in response to the shok. Look at the impulse response funtions below for the response to a suppl shok. The ke point is that the shok produes a persistent e et on in ation onl for values of near. 5

6 Again, this ma explain the di erene in eonomi performane between pre- and post-volker periods. 2. The Siene of Monetar Poli: A New Kenesian Perspetive CGG, JEL 999) Clarida, Gali and Gertler 999) surve man aspets of the monetar poli literature. This handout fouses upon the determination of optimal poli under various assumptions of the degree of redible ommitment b the entral bank. We will use the notation from the leture notes rather than the paper. The problem of the entral bank is to minimize its loss funtion subjet to the Phillips Curve: X h min E 0 t 2 i ) t + 2 t 9) s.t. t = E t f t+ g + t + u t 0) And the interest rate is given b substituting the solution to the above problem into the IS equation: n t+ o t = E t r t E t f t+ g rr t ) 2. Optimal Poli with Disretion This has been overed in the leture notes "Optimal Poli with a Cost-Push Shok," Setion.2. The entral bank has no abilit to in uene expetations about the future, so it takes the private setor s expetations as given in eah period. h min 2 i t + 2 t s.t. t = t + t where t E t f t+ g + u t is taken as given. As in the leture notes, the optimalit ondition for this problem is: t = t 2) 6

7 Let q 2 + u ). Then under the optimal time onsistent poli: t = qu t 3) 2.2 Classial In ation Bias t = qu t 4) The lassial in ation bias problem was formulated b Kdland and Presott 979) and Barro and Gordon 983). It onsiders the possibilit that the entral bank s target for the output gap is not 0 but k > 0. The entral bank solves: X min E 0 t t ) 2 k + 2 t Then the optimalit ondition under disretion is: s.t. t = E t f t+ g + t + u t k t = k t + k 5) where the supersript k highlights the fat that the objetive funtion has been amended to allow for k > 0. This ondition 5) an be used to haraterize the optimal output gap and in ation, then we ompare to the solution without a positive target for the output gap. k t = t 6) k t = t + k 7) The ke point is as follows. A entral bank with a positive output gap target does not in fat sueed in raising the output gap in equilibrium. Rather, the in ation simpl rises, b the amount k. Note that we have assumed =. A entral banker with a higher weight on in ation relative to output will have a lower, and hene a lower in ation bias term. That appointing a onservative entral banker an lead to welfare gains for soiet was proposed b Rogo 985). There are gains from inreasing redibilit. If ommitment b the entral bank is possible, then it will enhane welfare b allowing the entral bank to ommit to a response akin to the weight onservative entral bank. 2.3 Optimal Poli with Commitment But the abilit to ommit leads to welfare gains even if k = 0. This was illustrated in the leture notes in lass. I brie summarize the main results. n t o The monetar authorit hooses a state-ontingent poli ; t to solve: max 2 E X h 0 t 2 i ) t + 2 t The Lagrangean an be set up: s.t. t = E t f t+ g + t + u t for t = 0; ; 2; ::: max L = 2 E 0 X The solution obtained an be represented: t h 2 t + 2 t + t t t t+ i ) 0 = 0 9) t = t t for t = ; 2; 3; ::: 20) This is disussed in the leture notes. Intuitivel, there is histor dependene beause b ommitting to respond to the urrent shok partl toda and partl in the future, the tradeo between in ation and output is more favourable toda. 7

8 2.4 Optimal Poli within a Simple Famil of Poli Rules Now we allow an intermediate degree of ommitment, something that is not overed in the leture notes. To be spei, the ental bank an ommit to follow a poli where the output gap depends linearl upon the ontemporaneous shok term: t =!u t 2) Notie that suh a rule inludes the optimum under disretion as a speial ase. Therefore welfare must be weakl higher than in the disretion ase.) Combining equation 2) with the Phillips Curve implies that in ation under this rule is also linear in the ost-push shok: It is possible to express this as: t = E t t+ + t + u t X h = E t i i ) t+i + u t+i i=0 ) X = E t i [!u t+i + u t+i ] i=0 =! u u t t = t + u t 22) u u We now nd the optimal value of! b solving the following problem: max 2 E X t i 2 ) t+i + 2 t+i = 2 i=0 2 t + t ) 2 L t s.t. Equation 22) P where L t = E t i [u t+i =u t ] 2 > 0. i=0 The optimalit ondition is: t = t 23) where ) <. Sine <, ommitment takes the form of promising to engineer a greater ontration in output in response to in ationar shoks. There is a more aggressive response to prevent in ation from hanging muh from its target value. From a poli standpoint, Rogo s 985) point about a weight onservative entral banker arries over to this ase. The entral bank an seure a higher level of welfare if it promises to respond more vigorousl to an in ationar shok than the eonom s loss funtion would warrant. One wa in whih suh a ommitment ould be made is to appoint a entral banker with preferenes di erent from the general population, in partiular one that assigns more relative weight on ontrolling in ation rather than output. 8