– Form the Lagrangian 0 1 �Z n � 1 Z n (1 � � ) 0 c ( i ) � di � @ C h � b A � L = 0 p ( i ) c ( i ) di + � 1 First-order conditions are p ( i ) = �c ( i ) ( � � 1) b (1 � � ) C (1 � � ) 1 h which we use to write ! � � 1 p ( i ) c ( i ) = p ( j ) c ( j ) ! 1 p ( i ) � � 1 c ( i ) = c ( j ) p ( j )
– Use the expression for c ( i ) in the sub-index, 0 1 1 ! � �Z n � 1 � Z n (1 � � ) (1 � � ) p ( i ) � = b � � 1 B C 0 c ( i ) � di c ( j ) � di � � C h = b @ A 1 1 p ( j ) 0 �Z n � 1 (1 � � ) 1 � � 1 � � c ( j ) � � 1 di � = b p ( j ) 0 ( p ( i )) 1 �Z n � � 1 ( � � 1) 1 � � � � 1 di � ! c ( j ) = C h b p ( j ) 0 ( p ( i )) � � 1 1 – Now use the expression for c ( j ) in the budget constraint. Z n �Z n � � 1 Z n ( � � 1) � C h b � � � � 1 di � � 1 dj � P h C h = 0 p ( j ) c ( j ) dj = 0 ( p ( i )) 0 p ( j ) 1 �Z n � � 1 � Z n ( � � 1) � � � � 1 di � � 1 dj � P h = b 0 ( p ( i )) 0 p ( j ) 1 �Z n � 1 � 1 ( � � 1) � � � � 1 di � ! P h = b 0 ( p ( i )) 1
– Use the price index back in the c ( j ) expression to write the good’s demand �Z n � 1 � 1 � �Z n � � 1 ( � � 1) 1 � � � � 1 b � � 1 di � � 1 di � c ( j ) = p ( j ) 0 ( p ( i )) 0 ( p ( i )) C h 1 P h 1 = p ( j ) C h � � 1 � b 1 ( P h ) � � 1 ! 1 1 P h P 1 � � 1 1 � � = p ( j ) a C � � 1 1 � P h b 1 ( P h ) � � 1 1 ! 1 1 � � ! 1 1 � � c ( j ) = a P h P 1 � � 1 ! C p ( j ) b 1 P h
Reparameterization for Benigno’s model In Benigno’s JME paper, home consumer’s preferences for h 2 [0 ; n ] are 2 0 1 3 @ M h 1 � � � � X t + j U h � ( j � t ) C h L h A � V 4 U 5 t = E t + N t + j t + j P t + j j = t
where for elasticity of substitution between foreign and home � and elasticity of substitution across within country goods �; � � �= ( � � 1) n 1 =� C ( � � 1) =� � (1 � n ) 1 =� C ( � � 1) =� C = ; � > 0 H F "� 1 # �= ( � � 1) � 1 =� Z n 0 c ( z ) ( � � 1) =� dz C H = n "� # �= ( � � 1) � 1 =� Z 1 1 n c ( z ) ( � � 1) =� dz = C F 1 � n h i 1 = (1 � � ) nP 1 � � + (1 � n ) P 1 � � P = H F �� 1 � Z n � 1 = (1 � � ) 0 p ( z ) 1 � � dz P H = n "� # 1 = (1 � � ) � Z 1 1 n p ( z ) 1 � � dz P F = 1 � n Parameter correspondence:
� � 1 � = � 1 ! � = 1 � � � � 1 = 1 � � ! 1 � � ! � = � � 1 1 = � a 1 n For the general indices, h n 1 =� x ( � � 1) =� + (1 � n ) 1 =� y ( � � 1) =� i �= ( � � 1) 1 ( a 1 x � + a 2 y � ) � ! C = C = " # � � 1 � � 1 1 h i 1 = ( � � 1) � np 1 � � 1 � � � � 1 1 � � � � 1 + (1 � n ) p 1 � � = + a ! P = P a p p x y y 1 2 1 For sub-price indices,
� = ( � � 1) (1 � � ) ; ! 1 1 ( � � 1) = (1 � � ) 1 ! � = � = (1 � �: ) ! � � "� 1 # �= ( � � 1) �Z n � 1 � 1 =� Z n (1 � � ) � ! C H = 0 c ( i ) � di 0 c ( z ) ( � � 1) =� dz � = C h b 1 n �Z n � 1 � 1 �� 1 � Z n � 1 = (1 � � ) ( � � 1) � ! P H = � 0 p ( z ) 1 � � dz � � 1 di � P h = b 0 ( p ( i )) 1 n
3.1 Benigno’s model � Households – Notation � Home: [0 ; n ] : Foreign: ( n; 1] : Continuum of goods and agents. Each agent owns one …rm and supplies labor to that …rm. � h denotes home agent. f denotes foreign agent. � s t an event in period t: � s t = ( s t ; s t � 1 ; :::; s 0 ) is the history of events. � B h ( s t +1 ) : home agent’s holdings of one-period nominal state- contingent bonds.
� s t +1 j s t � � Q is the home-currency nominal price of the state-contingent bond. – Preferences: Home consumer: For h 2 [0 ; n ] ; 2 0 1 3 @ M h 1 � � � � X t + j U h � ( j � t ) C h L h 4 U A � V 5 t = E t + N t + j t + j P t + j j = t where for elasticity of substitution between foreign and home � and
elasticity of substitution across within country goods �; � � �= ( � � 1) n 1 =� C ( � � 1) =� � (1 � n ) 1 =� C ( � � 1) =� C = ; � > 0 H F "� 1 # �= ( � � 1) � 1 =� Z n 0 c ( z ) ( � � 1) =� dz C H = n "� # �= ( � � 1) � 1 =� Z 1 1 n c ( z ) ( � � 1) =� dz = C F 1 � n h i 1 = (1 � � ) nP 1 � � + (1 � n ) P 1 � � P = H F �� 1 � Z n � 1 = (1 � � ) 0 p ( z ) 1 � � dz P H = n "� # 1 = (1 � � ) � Z 1 1 n p ( z ) 1 � � dz P F = 1 � n
– Budget constraint: Income comes from pro…ts of …rms owned by households, payo¤s from state-contingent bonds, wages, and trans- fers from the government. � s t +1 j s t � B h ( s t +1 ) = B h ( s t ) + M h + M h + W h t L h + � h + Tr h Q X t � 1 C h t t t t t + P t P t P t P t P t P t P t s t +1 – Government budget constraint: Transfers are helicopter drops of cash. Z n Z n � � M h t � M h 0 Tr h dh = t dh t � 1 0
– For the intertemporal allocation problem, write down two terms from the Lagrangian � s t � 2 0 1 3 @ M h � s t ; s t � 1 � � � s t �� � � s t �� � t X t C h L h 4 U A � V 5 � + N t t P t ( s t ) s t 0 1 � s t � Q ( s t +1 j s t ) B h ( s t +1 ) + M h t ( s t ) + P C h � s t � B C s t +1 X t P t ( s t ) P t ( p t ) B " # C � � t B C t � 1 ( s t � 1 ) + W h B h ( s t )+ M h t ( s t ) L h t ( s t ) + � h t ( s t ) + Tr h t ( s t ) @ A � s t P t ( s t ) � s t +1 � 2 0 1 3 @ M h � s t +1 � � � s t +1 �� � � s t +1 �� � t +1 X t +1 C h L h A � V 4 U 5 � � � + N t +1 t +1 s t +1 P t +1 s t +1 0 � s t +1 � Q ( s t +2 j s t +1 + P C h B � s t +1 � � s t +1 � s t +2 X t +1 P t +1 B " � � � t +1 B M h t +1 ( s t +1 ) B h ( s t +1 )+ M h t ( s t ) + W h t +1 ( s t +1 ) L h t +1 ( @ s t +1 + P t +1 ( s t +1 ) � P t +1 ( s t +1
� Obtain the …rst-order conditions for consumption, � s t ; s t � 1 � U 0 � � s t �� � s t � C h � t � C h : = � t t t � s t +1 j s t � � s t � Q 1 B h ( s t +1 ) : � t = � t +1 ( s t +1 ) P t ( s t ) P t +1 ( s t +1 ) Eliminating the multiplier gives the price of state contingent bonds s t +1 j s t � U 0 � � s t +1 �� C h � s t +1 j s t � � P t t +1 � � Q = �� C h U 0 P t +1 t from which it follows that the price of one-period riskless nominal bonds are U 0 � � 8 9 C h � s t +1 j s t � < = X 1 P t t +1 � � = Q = �E t : ; C h 1 + i t U 0 P t +1 s t +1 t
� Euler equations for labor and real money holdings: � s t � s t �� W h � s t � = U 0 � � t C h L : ! V L t P t ( s t ) ! = U 0 � � s t �� M t i t C h : M ! N M t P t 1 + i t � In equilibrium, we can drop the h superscript on consumption. Due to complete markets, everyone’s consumption will be equal. � Home consumer demand for the home good and the foreign good ! � � � P H � � � p ( h ) c h ( h ) = C P H P ! � � � P F � � � p ( f ) c h ( f ) = C PF P with similar demand functions for the foreign guy.
� Complete international asset markets. One-period state contingent nom- inal bonds denominated in the home currency are traded internationally. Home guy’s Euler equation has already been found. For foreign guy, � s t � U 0 � � s t �� � s t � C f C f = � f � t � : t t t � s t +1 j s t � � s t � Q 1 B h ( s t +1 ) � f : = � t +1 ( s t +1 ) t S t P � S t P � t ( s t ) t +1 ( s t +1 ) s t +1 j s t � U 0 � � s t +1 �� � s t � C f S t P � � s t +1 j s t � � t t +1 Q = �� � � S t +1 P � C f t +1 ( s t +1 ) U 0 t ( s t ) s t +1 j s t � U 0 � � s t +1 �� C h � P t t +1 � � = �� C h U 0 P t +1 t
� By the law of one price, U 0 � � s t +1 �� U 0 � � C f C h S t P � P t +1 t +1 t t � � = � � �� S t +1 P � C f C h P t s t +1 U 0 U 0 t +1 t t +1 and after backwards recursive substitution, U 0 � � C f � = �S t P � t t � = �RS t C h U 0 P t t where � depends on initial conditions. This is the Backus–Smith con- dition. Benigno achieves real exchange rate dynamics through relative movements in consumption.
� Monetary policy is conducted through interest-rate feedback rules (Taylor t be monetary policy shocks, F and F � be a set of Let " H t , " F rules). target variables. The general speci…cation of the rules are � � 1 + i t F; s t ; " H = � t 1 + � { 1 + i � � � � t F � ; s t ; " F = t { � 1 + � 8 9 U 0 � � U 0 � � 8 9 C f > > C h < = < = P � P t t +1 t +1 ; = (1 + i � t � � 1 = (1 + i t ) �E t t ) �E t � � P � : > C f > C h U 0 P t +1 U 0 : ; t +1 t t
� Production with local currency pricing. All produced goods are trad- able. Deviations from PPP are due to international market segmentation. Prices are sticky in terms of the local currency. – Sticky prices through the Calvo mechanism. Say the Calvo-lottery chooses home …rm h to set a new price at time t: The …rm does so to maximize expected present value of net pro…ts. Let � s be the probability that the price will remain in e¤ect from t + 1 through t + s: The owner’s discount factor is � t;t + s = � s U 0 ( C t + s ) P t U 0 ( C t ) P t + s and the production technology t ( h ) + c f c h t ( h ) = y t ( h ) = A t L h t where A t is a country-speci…c technology (productivity) shock.
– The objective is to maximize 2 3 1 X 6 7 t + j ( h ) y f � j � t;t + j 4 p t + j ( h ) y h + S t + j p � � W h t + j L h 6 7 E t t + j ( h ) t + j ( h ) 5 t + j | {z } | {z } | {z } j =0 home sales wage bill foreign sales subject to ! � � " P H t # � � p t ( h ) y h nC t = � h t p t ( h ) � � t ( h ) = P Ht P t 0 1 � � " P � # � � @ p � t ( h ) y f H;t (1 � n ) C � t = � � t p � t ( h ) � � A t ( h ) = P � P � t H t where ! � � " P H t # � � 1 � h = nC t t P Ht P t 0 1 � � " P � # � � @ 1 H;t � � (1 � n ) C � A = t t P � P � t H t
The …rm must set p t ( h ) and p � t ( h ) . h i t p t ( h ) � � + S t p � t ( h ) � � � W h p t ( h ) � h t ( h ) � � t p � t L h � t;t t � t ( h ) � � � t p t ( h ) � � � � � A t L h t � � h t p � + � t h t +1 p t +1 ( h ) � � + S t +1 p � t +1 ( h ) � � � W h p t +1 ( h ) � h t +1 ( h ) � � t +1 p � + �E t � t;t +1 t +1 L � t +1 ( h ) � � � t +1 p t +1 ( h ) � � � � � A t +1 L h t +1 � � h t +1 p � + E t � t +1 + � � � � First, choose p t ( h ) ; with the expectation that it will stay in e¤ect. 1 1 X X p t ( h ) : (1 � � ) p t ( h ) � � E t t + j � �p t ( h ) � (1+ � ) E t � j � t;t + j � h � h t + j j =0 j =0 and rearranging gives P 1 j =0 � h E t � t + j p t ( h ) = P 1 j =0 � j � t;t + j � h (1 � � ) E t t + j
� Similarly, the optimal way to set the foreign price is, P 1 j =0 S t + j � � E t � t + j p � t ( h ) = P 1 j =0 � j � t;t + j � � (1 � � ) E t t + j � Evolution of price sub-indices. A measure � of the …rms have to keep the price as before. (1 � � ) set to the new price p t ( h ) ; therefore, P H;t = � h P H;t � 1 + (1 � � h ) p t ( h ) P � � � h P � H;t � 1 + (1 � � � h ) p � = t ( h ) H;t with similar equations holding for the foreign …rm � � P � � � f P � 1 � � � p � = F;t � 1 + t ( f ) F;t f � � P F;t = � f P F;t � 1 + 1 � � f p t ( f ) Benigno allows for the possibility that the degree of price stickiness may di¤er across countries.
� Relative price indices of the imported good in terms of the domes- tically produced good in terms of the local currency P F T = P H P � T � H = P � F
� Aggregate ‡uctuations. – Expand around a steady state with zero in‡ation and exchange rate depreciation, nominal interest rates equal to utility-based discount rate, and a real exchange rate of 1 and equal consumption across countries. – Highlights of the log-linearization. 1. � = ln (1 + � ) 2. d RS t = ln ( RS t ) � P � � � P F � T � = ln 3. ^ ; ^ H T = ln P � P H F 4. ~ C; ~ T are ‡exible-price equilibrium values. 5. � = LV LL V L ; � = � CU CC U c
– Relative prices and the real exchange rate have unit roots. Obtain an IS-like curve, uncovered interest parity, and an expenditure switching e¤ect. ^ ^ T t = T t � 1 + � F;t � � H;t T � T � ^ ^ = t � � F;t + � H;t t RS t � 1 + � � d d RS t = t � � t + � S t C t + 1 E t ^ ^ C t +1 = � [^ { t � E t � t +1 ] { � E t � S t +1 = ^ { t � ^ t � ^ � ^ � � �� T � T � T t � ~ t � ~ y H;t � y F;t = � n T t � (1 � n ) t
– Dynamics under sticky prices � � h = � � h = � f = � � f : Chari-Kehoe-McGrattan case. � � h = � f ; � � h = � � f : Location of consumption determines degree of price stickiness. � � h = � � � f = � � : h ; f : Price stickiness is …rm location speci…c and faces the same stickiness in the home and foreign market. � Monetary policy rules re‡ect interest rate smoothing and exchange rate feedback. { t � 1 + � H � t + H y H t + � H � S t + � H d RS t + " H ^ { t = � H ^ t { � { � t � 1 + � F � � t + F y F RS t + " F t � � F � S t � � F d ^ = � F ^ t t " j t = � j " j t � 1 + v j;t
� Alternative policy rules – Fixed exchange rates. Foreign country equates i � to i with a reaction � � to deviations of the exchange rate from target. Let ^ S= � S = ln S , where � S is the target. home country follows Taylor rule { � { t � � ^ ^ = ^ S t t ^ = �� t + y H;t { t – Taylor rules ^ { t = �� t + y H;t + ^ " H;t { � �� � ^ = t + y F;t + " F;t t – Managed ‡oat ^ { t = �� t + y H;t { � �� � ^ = t + y F;t � � � S t t
� Tradeo¤ between persistence and volatility as response to exchange rate � increases.
� Interest rate smoothing induces additional inertia. � is the smootheing coe¢cient. For symmetric Taylor rules
� Preliminary results – Suppose � H = � � H = � F = � � F : � Then price dynamics are synchronized and the relative prices T = P F =P H and T � = P � H =P � F are uncorrelated with monetary policy. Relative prices will be a¤ected only by productivity shocks. � The real exchange rate displays no persistence following a monetary shock under in‡ation targeting or under the Taylor rule. � After a monetary shock, the nominal and real exchange rates return to equilibrium after one period, as in the Redux model. � There will be persistence in the exchange rates if monetary shocks are serially correlated. � Under in‡ation targeting, the real exchange rate is isolated from productivity shocks.
� When there is no weight on output stabilization, productivity shocks wil have no e¤ect on the real exchange rate. Also, the in‡ation di¤erential is independent of relative price changes. The link between productivity shocks and the real exchange rate is broken.
� Plausible examples of persistence generation in the real exchange rate: Calibration and simulations. � = 0 : 99 ; � = 2 ; n = 0 : 5 ; � = 6 ; � = 10 ; � = 1 : 5 : The only shock is from domestic monetary policy with no autoregressive component.
– Interest rate smoothing
– Firm-speci…c price rigidity � H = � � H 6 = � F = � � F
H = 0 : 66 ; � F = 0 : 67 ; � � – � H = 0 : 8 ; � � F = 0 : 4 : , � = 0 : 85 ; � = 0 : 225
� Takeaway: Previously thought: Persistence of real exchange rate increas- ing in degree of price stickiness, and unrealistically long period of price stickiness to match the persistence in real exchange rate. Now thought: Price rigidity is not su¢cient by itself to generate persistence following a monetary shock.
4 In‡ation News and the Exchange rate: Clar- ida and Waldman Theoretical model for how bad news about in‡ation (higher than expected) is good news for the exchange rate (home currency strengthens). Interesting and nicely constructed empirical work. Questions: 1. What is the correlation between in‡ation surprises and changes in the nominal exchange rate? 2. Is the sign di¤erent for in‡ation targeters and non-in‡ation targeters?
� Data: 5-minute nominal spot exchange rate data for USD–JPY,CAD,NOK,SEK,CHF,EUR,GB and NZD, July 2001 through December 2005. � Construct returns 10-minute percentage changes to capture behavior in a plus/minus 5 minute window of the announcement. � Positive in‡ation surprise means higher than expected. Expectations is the median survey response from Bloomberg News Service, which surveys commercial and investment banks on macroeconomic announcements. Use consumer price in‡ation. Month-over-month (MoM) and year-over- year (YoY) in‡ation for headline and core in‡ation. � Regression. R t is 10 minute return around the announcement. R > 0 means home currency appreciates . S t is the in‡ation surprise. R it = � + �S it + u it
– Normalize coe¢cients to interpret � as an elasticity. – Pool the data and run as stacked OLS regression.
– In‡ation targeters versus non-in‡ation targeters.
– Regime changes: Bank of England independence (1997), Norway’s formal adoption of in‡ation targeting (2001). Before the regime change, coe¢cients are negative (not signi…cant).
5 Calibrated partial equilibrium models of the exchange rate and Taylor rules Engel and West JMCB. Rational expectations model. Mark (mimeo). Adaptive learning
� Partial equilibrium setup. We assume – The public believes Fed and Bundesbank use some form of the Taylor rule. – Public views in‡ation and output gap are exogenously generated by a VAR(4). – Economic model is real interest parity, a stochastic di¤erence equa- tion, whose solution gives the real exchange rate. – Public employs least-squares learning rules to form beliefs about model’s coe¢cients since their true values are unknown and may change over time. � Feed historically observed data on in‡ation and the output gap into the model. Observe the implied learning equilibrium path of the real ex- change rate and compare with historical real exchange rate path.
5.1 Can we assume that the Fed and Bundesbank (ECB) reacts the same to in‡ation and the output gap? � German variables subscripted by ‘G’ U.S. variables subscripted by ‘U.’ � German–U.S. di¤erentials have no special notation. � � � t = � G;t � � U;t � � i t = i G;t � i U;t � � x t = x G;t � x U;t are German-U.S. di¤erentials in in‡ation, short-term nominal interest rates and activity gaps, respectively. Activity gap de…ned such that the economy operates in excess of its potential when x tj > 0 ; j = G; U: The log real exchange rate is q t .
� Fed rule for target rate � � i T U;t = i U + � � + � x x U;t : E t � U;t +1 � � U Actual interest rate subject to exogenous and i.i.d. policy shock � U;t and interest rate smoothing i U;t = (1 � � ) i T U;t + �i U;t � 1 + � U;t : � Bundesbank target rule � � i T G;t = � { G + � � E t � G;t +1 � � G + � x x G:t + � q q t � 1 ;
� Impose homogeneity of the coe¢cients ( � � ; � x ) across countries and write interest di¤erential as � � i t = � + �i t � 1 + (1 � � ) � � E t � t +1 + � x x t + � q q t � 1 ; + � t ; �� � � iid � (0 ; � 2 – � � (1 � � ) i G � i U � � � ( � G � � U ) ; and � t � ) : – Add and subtract (1 � � ) � � e � t +1 on the right side of and rearrange i t = � + (1 � � ) [ � � � t +1 + � x x t + � s q t � 1 ] + �i t � 1 + � 0 t ; where � 0 t = � t � (1 � � ) � � [ � t +1 � E t � t +1 ] : is uncorrelated with date t information. Estimate by GMM. Instru- ments are a constant, the current value and three lags of the in‡ation di¤erential, the current value and three lags of the output (alterna- tively unemployment) gap di¤erential, four lags of the nominal inter- est di¤erential, and four lags of the real exchange rate.
Table 2: GMM Estimates of Bundesbank–Fed Relative Interest-Rate Reaction Function with Lagged Real Exchange Rate Feedback. Bold indicates signi…cance at the 5% level Source output gap Sample � � � � � x � q J-statistic (t-ratio) (t-ratio) (t-ratio) (t-ratio) (t-ratio) (p-valu 61.2-79.2 -0.006 0.666 0.034 0.134 0.019 9.684 (-6.477) (10.545) (0.376) (5.092) (4.069) (0.644) 79.3-05.4 -0.001 0.895 1.318 0.376 0.011 4.330 (-1.194) (22.450) (3.063) (2.318) (0.880) (0.977) Structural change test All coe¤s. Test statistic 40.037 0.000 In‡ation coe¤. Test statistic 8.525 0.004
HP output gap � � � � � x � q J-statistic (t-ratio) (t-ratio) (t-ratio) (t-ratio) (t-ratio) (p-valu 61.2-79.2 -0.003 0.745 0.121 0.045 0.011 9.370 -3.589 12.551 1.357 1.166 2.708 0.671 79.3-05.4 0.000 0.873 1.544 0.428 0.004 5.314 -0.202 24.480 4.406 3.239 0.558 0.947 Structural change test All coe¤s. Test statistic 36.747 0.000 In‡ation coe¤. Test statistic 15.501 0.000
HP unemployment gap J-statistic � � � � � x � q (t-ratio) (t-ratio) (t-ratio) (t-ratio) (t-ratio) (p-valu 61.2-79.2 -0.003 0.579 0.034 -0.644 0.005 7.746 -3.077 8.394 0.554 -4.847 1.873 0.805 79.3-05.4 0.000 0.860 1.154 -0.828 0.004 5.083 -0.163 22.028 2.812 -3.171 0.443 0.955 Structural change test All coe¤s. Test statistic 25.042 0.015 In‡ation coe¤. Test statistic 7.280 0.007
6 Modeling real exchange rate dynamics with learning � Economic model is uncovered interest parity. For the log nominal ex- change rate s t (DM/$) s t = E t s t +1 � i t : where i is the German–US interest di¤erential. � Add and subtract E t � t +1 from the right side, rearrange q t = E t q t +1 � i t + E t � t +1 � � i t = � + �i t � 1 + (1 � � ) � � E t � t +1 + � x x t + � q q t � 1 ; + � t
� Let � and x be exogenously given by VAR(4). Let Y 0 t = ( � t ; : : : ; � t � 3 ; x t ; : : : ; x t � 3 ) ; � : Regression form of the VAR is � Y 0 and Z 0 1 ;t = Z 0 2 ;t = t ; 1 B 0 1 Z 0 � t = 1 ;t � 1 + v 1 t ; B 0 2 Z 0 x t = 2 ;t � 1 + v 2 t ; B 1 and B 2 are 9 x 1 least-squares coe¢cient vectors. � Companion representation Y t = � + AY t � 1 + v t ; e 1 = (1 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0) � t = e 1 Y t e 2 = (0 ; 0 ; 0 ; 0 ; 1 ; 0 ; 0 ; 0) x t = e 2 Y t E t � t +1 = e 1 ( � + AY t ) :
� The above gives a second-order stochastic di¤erence equation in q t , q t = � ((1 � � ) � � � 1) e 1 � + (1 � � ) � q q t � 1 + �i t � 1 + ((1 � � ) � x e 2 + ((1 � � ) � � � 1) e 1 A ) Y t + � t + E t q t +1 � Rational expectations solution q t = a 0 + a 1 i t � 1 + a 2 q t � 1 + a 3 � t + bY t q (1 � � ) 2 � (1 � � ) 4 � q 1 a 2 = 2 (1 � � ) � ; 2 � �a 2 a 1 = ; (1 � � ) � q ! (( � 1 + (( � � 1) a 1 + (1 � � )) � � ) e 1 � b ) � � ( a 1 � 1 ) � a 0 = ; a 2 a 2 1 � a 1 a 3 = ; 1 � a 2 (((1 + ( a 1 � 1) (1 � � ) � � ) e 1 ) A + ( a 1 � 1) (1 � � ) � x e 2 ) ((1 � a 2 ) I � A ) � 1 : b =
– Non-uniqueness. Choose solution with positive a 2 – b depends on in‡ation response coe¢cient � � . � � � < 1 in pre-Volker sample: Decline in E t � t +1 leadds public to expect increase in the r and real dollar depreciation. � � � > 1 in the post-1979 sample: Decline in E t � t +1 leads to real dollar appreciation
� Learning the rational expectations equilibrium. Solve real interest parity condition using expectations formed from perceived law of motion. – At t , coe¢cient vectors B 1 ;t � 1 ; B 2 ;t � 1 are given (estimated in t � 1) : From regression form, B 0 � t = 1 ;t � 1 Z 1 ;t + u 1 ;t ; B 0 x t = 2 ;t � 1 Z 2 ;t + u 2 ;t ; form companion form Y t = � t � 1 + A t � 1 Y t � 1 + v t ; Construct expected in‡ation E t � t +1 = e 1 ( � t � 1 + A t � 1 Y t ) : – Believe the rational expectation solution and use it for perceived law of motion q t = a 0 ;t � 1 + a 1 ;t � 1 i t � 1 + a 2 ;t � 1 q t � 1 + a 3 ;t � 1 � t + b t � 1 Y t � B 0 3 ;t � 1 Z 3 ;t
– Observe the policy shock � t from perceived law of motion for the interest di¤erential = � t � 1 + � t � 1 i t � 1 + � �;t � 1 e 1 ( � t � 1 + A t � 1 Y t ) + � x;t � 1 e 2 Y t + � q;t � 1 q t � 1 + � i t B 0 � 4 ;t � 1 Z 4 ;t + � t : – The expected exchange rate E t q t +1 = a 0 ;t � 1 + a 1 ;t � 1 i t + a 2 ;t � 1 q t + b t � 1 ( � t � 1 + A t � 1 Y t ) : – Plug in‡ation forecast and q forecast into real UIP gives the actual law of motion, h � � i 1 � � q t = a 0 ;t � 1 + ( e 1 + b t � 1 ) ( � t � 1 + A t � 1 Y t ) + a 2 ;t � 1 � 1 i t : 1 � a 3 ;t � 1 – For next period, least-squares update the coe¢cients Let y 1 ;t = � t ; y 2 ;t = x t; y 3 ;t = q t ; and y 4 ;t = i t : For given R j;t � 1 ( j = 1 ; :::; 4) ;
and a …xed gain g; the updating formulae are � � Z j;t � 1 Z 0 = R j;t � 1 + g (1) R j;t j;t � 1 � R j;t � 1 ; B j;t � 1 + gR � 1 j;t Z j;t � 1 ( y j;t � B 0 = j;t � 1 Z j;t � 1 ) : (2) B j;t The learning path and coe¢cient updating is generated using ob- servations of � t ; x t ; and i t from the data, but not with exchange rate data. The learning values of q t generated by the actual law of motion and employed in coe¢cient updating are generated solely as functions of �; x; and i .
� Calibrated Learning and Rational Exchange Rate Paths The observations are standardized to highlight comovements between the series.
Figure 5. Learning path with source output gap.
The learning path shown in Figure 5 generally captures closer comovements with the data and does a good job of capturing the dollar cycle of the 1980s. The learning path exhibits the dollar appreciation and subsequent depreciation in the latter part of the sample although timing of the turning points are o¤ a bit with the phase of the learning cycle leading the data.
Figure 7. Learning path with HP output gap.
The learning path in Figure 7 shows the dollar prematurely appreciating in 1979.2 and it does not generate quite the strength attained in the data by 1984.4. The learning path also leads the turning points in the dollar appre- ciation and depreciation beginning in 1995.2 but otherwise comoves with the data.
Figure 9. Learning path with HP unemployment gap.
Figure 9, it is seen that the comovmements of the learning path with the data are also quite good. The learning path matches the timing of the 1980s dollar cycle very well. Except for a phase shift that leads the data, it also captures the cycle from 1995 to 2005
Figure 4: Rational path with source output gap.
Using the source output gap, the rational path shown in Figure 4 misses a good deal of the real dollar appreciation from 1980.4 to 1984.4 and falsely predicts a dollar appreciation from 1989.3 to 1991.2. It also erroneously predicts a large dollar depreciation in 1981.1 Apart from these episodes, there is a close connection between the implied rational expectations path and the data. The rational expectations path does a good job of explaining the real dollar appreciation from 1996.2 to 2002.2 and the subsequent real dollar depreciation.
Figure 6: Rational path with HP output gap.
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