Introduction to Dynamic Models. Slide set #1 (Ch in IDM).

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1 1 Introduction Introduction to Dynamic Models. Slide set #1 (Ch in IDM). Ragnar Nymoen University of Oslo, Department of Economics We observe that economic agents take time to adjust their behaviour to changes in circumstances. Instantaneous adjustment is the exception in economics. Adjustment lag is the rule. Dynamic behaviour is therefore pervasive in economics. Models with a dynamic formulation are therefore needed. And we need a methodology for understanding and analyzing dynamic models. January 13, The importance of dynamics is recognized by policy decision makers:. Norges Bank [The Norwegian Central Bank] is typical of many central banks view: A substantial share of the effects on inflation of an interest rate change will occur within two years. Two years is therefore a reasonable time horizon for achieving the inflation target of per cent1 One important aim of this course is to learn enough about dynamic modeling to be able to understand the economic meaning of a statement like this, and to start forming an opinion about its realism (or lack thereof). 1 policy/in norway.html. Similar statements can be found on the web pages of the central banks in e.g., Autralia, New-Zealand, The United Kingdom and Sweden. 3 Dynamic models often include both flow and stock variables. Flow: in unit of (for example) million kroner per year Stock: in units of (for example) million of kroner at a particular period in time (for example start or end of the year). In this course an important class of stock variables will be price indices. For example P t may represent the value of the Norwegian CPI in period t (a month, a quarter or a year). As you know, the values of P will be index numbers. The number 100 (often 1 is used instead) refers to the base period of the index. If P t > 100 it means that relative to the base period, prices are higher in period t. 4

2 600 CPI_Norway CPI_UK Starting from a stock variable like P t,aflow variable results from obtaining the change, hence x t = P t P t 1, the (absolute) change y t = P t P t 1, P t 1 the relative change, and z t =lnp t ln P t 1 the approximate relative change are examples of flow variables. Note that: Inflation_Norway Inflation_UK y t 100 is inflation in percentage points. In this course we often stick to the rate formulation (hence, we omit the scaling by 100) z t y t by the properties of the (natural) logarithmic function, see for example the appendix of IDM, if in doubt Figure 1: Consumer price indices (stock variables), and their rate of change (flow). Norway and UK 5 6 An typical empirical trait of stock variables are that they change gradually, as a result of finite growth rates. Sometimes however, stock variables jump from one value in period t to quite another in period t +1. Empirically, the rate of change then becomes very large. Norwegian price history at the breakdown of the union with Denmark is an example. In economics, when stock variables change gradually, we need explicit dynamic models to account for their evolution. Sometimes though, stock variables can be treated theoretically as if the are jump-variables. An example of such a theory in this course is the portfolio model of the foreign exchange market. That model is static. In the light of the pervasiveness of dynamics in real world phenomena: what is the rationale and interpetation of static model? 1. as a first approximation to actual behaviour, when the speed of adjustment is fast (although a dynamic model would give better understanding). The portfolio model is an example. 2. as steady-state relationships, derived from and consistent with dynamic models. We will use both interpretations in our course, and need to distinguish between them. Developing the 2. interpretation in a key goal in the firstpartofthecourse. 7 8

3 75 The Norwegian current account As noted economic dynamics often arise from the combination of flow and stock variables. For example, the dynamic behaviour of debt (a stock) is linked to the value of the current account (flow) in the following way debt = current account + last periods debt + corrections. For example: If there is a primary account surplus for some time (and ignoring corrections for simplicity), this will lead to a gradual reduction of debt or an increase in the nation s net wealth. Conversely, a consistent current account deficit raises a nation s debt. Billion kroner Billion kroner Norwegian net foreign debt Figure 2: The Norwegian current account (upper panel) and net foreign deb (lower panel)t. Quarterly data 1980(1)-2003(4) Static and dynamic models an example In the lecture notes, Introductory Dynamic Macroeconomics (IDM) further examples of dynamics are given in Chapter 1.1: Capital stock dynamics (economic growth) The textbook consumption function, i.e., the relationship between real private consumption expenditure (C) and real households disposable income (INC) is an example of a static equation C t = f(inc t ), f 0 > 0. (1) Two examples of specified functional forms for the static consumption function: Dynamics of consumption and income (flows) and private household wealth. Chapter 1.2 provides a detailed example (see below) C t = β 0 + β 1 INC t + e t, (linear) ln C t = β 0 + β 1 ln INC t + e t, (log-linear) (2) If this is unfamiliar, read in IDM about the properties of these two functional forms.forexamplefortheinterpretationofβ. Textbooks usually omit the term e t in equation (2), but for applications of the theory to actual data it is a necessary to get an intuitive grip on this disturbance term in the static consumption function

4 12.0 Using quarterly data for Norway, for the period 1967(1)-2002(4), we obtain by using the method of least squares method: ln Ĉ t = ln INC t (3) where the hat in Ĉ t is used to symbolize the fitted value. Next, use (2) and (3) to define the residual ê t : which is the empirical counterpart to e t. ê t =lnc t ln Ĉ t, (4) ln C t ln INC t Figure 3: The estimated static consumption function The dynamic consumption function: ln C t = β 0 + β 1 ln INC t + β 2 ln INC t 1 + α ln C t 1 + ε t (5) is an example of a so called autoregressive distributed lag model, ADL. Estimated: ln Ĉ t = ln INC t ln INC t ln C t 1 (6) Compare residuals ˆε t and ê t to judge which model is best (see graph). Which explanatory variables contribute most to the improved fit? ln C t 1 itself! Illustrates that the dynamic framework is important Residuals of (2.7) Residuals of (2.4) The rather low values of the income elasticities (0.130 and 0.08) reflect that a single quarterly change in income is too little to build on. We will see that the results imply a much higher impact of a permanent change in income than of a temporary rise Figure 4: Residuals of the two estimated consumptions functions (3), and (6), 15 16

5 3 Dynamic multipliers A substantial share of the effects on inflation of an interest rate change will occur within two years. Two years is therefore a reasonable time horizon for achieving the inflation target of per cent This may mean that the effect is building up gradually over 8 quarters and then dies away quite quickly, but other interpretations are also possible. In order to inform the public more fully about its view on the monetary policy transmission mechanism (see topic 5 in our course), the Bank would have to give a more detailed picture of the dynamic effects of a change in the interest rate. To make progress we need to understand fully a concept called the dynamic multiplier. In order to explain dynamic multipliers, we first show what our estimated consumption function implies about the dynamic effect of a change in income (Ch 1.3 of IDM). Next, we derive the dynamic multipliers using a general notation, see Ch 1.4 of IDM Dynamic effects of income on consumption Simplify equation (6) by setting ε t = 0, hence we can drop the ˆ above C t. Assume that income rises by 1% in period t, so instead of INC t we have INC 0 t = INC t( ). Using (6), we have ln(c t (1+δ c,0 )= ln(inc t (1+0.01))+0.08 ln INC t ln C t 1 where δ c,0 denotes the relative increase in consumption in period t, thefirst period of the income increase. Since ln(1 + δ c,0 ) δ c,0 when 1 <δ c,0 < 1, and noting that ln C t ln INC t 0.08 ln INC t ln C t 1 =0 we obtain δ c,0 = , meaning that in percentage terms the immediate effect is a 0.13% rise in consumption. 18 What about the second period after the shock? Note first that the estimated model also holds for period t + 1, hence ln(c t+1 (1 + δ c,1 )) = ln(inc t+1 ( )) +0.08(ln INC t ( )) ln(c t (1 + δ c,0 ), after the shock. The relative increase in C t in period t + 1 becomes δ c,1 = = , or 0.3%. By following the same way of reasoning, we find that the percentage increase in consumption in period t +2is0.46% (formally δ c,2 100). Since δ c,0 measures the direct effect of a change in INC, it is usually called the impact multiplier, and is defined by taking the partial derivative ln C t / ln INC t in equation (6). The dynamic multipliers δ c,1,δ c,2,...δ c, are in their turn linked by the dynamics of equation (6), namely For example, for j =3,andsettingδ inc,3 = δ inc, =0.01, a permanent rise in income,weobtain δ c,3 = = or 0.57% in percentage terms. The long-run multiplier: Set δ c,j = δ c,j 1 = δ c,long run we obtain δ c,long run = =0.01, a 1% permanent increase in income has a 1% long-run effect on consumption. δ c,j =0.13δ inc,j +0.08δ inc,j δ c,j 1,forj =1, 2,.... (7) 19 20

6 1.00 Temporary change in income 1.10 Permanent change in income Permanent 1% change Temporary 1% change Impact period period after shock period after shock long-run multiplier Percentage change Period Dynamic consumtion multipliers (temporary change in income) Percentage change Period Table 2: Dynamic multipliers of the estimated consumption function in (6), percentage change in consumption after a 1 percent rise in income. Percentage change 0.05 Percentage change Dynamic consumption multipliers (permanent change in income) Period Period Figure 5: Temporary and permanent 1 percent changes in income with associated dynamic multipliers of the consumption function in (6) The distinction between short and long-run multipliers permeates modern macroeconomics, and so is not special to the consumption function! B&W: Chapter8,onmoneydemand,Table8.4. Chapter 12, where short and long-run supply curves are derived. For example, the slopes of the short-run curves in figure 12.6 correspond to the impact multipliers of the respective models, while the vertical long-run curve suggest that the long-run multipliers are infinite (we ll return to this) Norges Bank on inflation targeting and many, more examples. 4 General notation of the ADL model ADL model: y t is the endogenous variable while the x t and x t 1 make up the distributed lag part of the model: y t = β 0 + β 1 x t + β 2 x t 1 + αy t 1 + ε t. (8) Define x t, x t+1, x t+2,,... as functions of a continuous variable h. When h changes permanently, starting in period t: x t / h > 0. Since x t is a function of h, soisy t,andtheeffect of y t of the change in h is found as y t h = β x t 1 h. It is customary to consider unit changes in the explanatory variable, which means that we let x t / h = 1. Hence the first multiplier is y t h = β 1. (9) 23 24

7 The second multiplier is found by considering the equation for period t +1, i.e., y t+1 = β 0 + β 1 x t+1 + β 2 x t + αy t + ε t+1. and calculating the derivative y t+1 / h : y t+1 h = β x t+1 1 h + β x t 2 h + α y t (10) h Again, considering a unit change, and using (9), y t+1 h = β 1 + β 2 + αβ 1 = β 1 (1 + α)+β 2 (11) The pattern in (10) repeats itself for higher order multipliers, hence: multiplier number j +1isgivenas whereweusethenotation: δ j = β 1 + β 2 + αδ j 1,forj =1, 2, 3,... (12) δ j = y t+j, j =1, 2,... h 25 The long run multiplier is defined by setting δ j = δ j 1 = δ long run. Using (12), δ long run is found to be δ long run = β 1 + β 2,if 1 <α<1. (13) 1 α Clearly, if α = 1, the expression does not make sense mathematically, since the denominator is zero. Economically, it doesn t make sense either since the long run effect of a permanent unit change in x is an infinitely large increase in y (if β 1 + β 2 > 0). The case of α = 1, may at first sight seem to be acceptable since the denominator is 2, not zero. However, as explained below, the dynamics is essentially unstable also in this case meaning that the long run multiplier is not well definedforthecaseofα = Table 3: model. Dynamic multipliers of the general autoregressive distributed lag 5 A typology of linear models ADL model: y t = β 0 + β 1 x t + β 2 x t 1 + αy t 1 + ε t. Permanent change in x (1) Temporary change in x (2) 1. multiplier: δ 0 = β 1 δ 0 = β 1 2. multiplier: δ 1 = β 1 + β 2 + αδ 0 δ 1 = β 2 + αδ 0 3. multiplier:. δ 2 = β 1 + β 2 + αδ 1. δ 2 = αδ 1. j+1 multiplier δ j = β 1 + β 2 + αδ j 1 δ j = αδ j 1 The discussion at the end of the last section suggests that if the coefficient α in the ADL model is restricted to for example 1 or to 0, quite different dynamic behaviour of y t is implied. In fact the resulting models are special cases of the unrestricted ADL model. For reference, this section gives a typology of models that are encompassed by the ADL model. Some of these model we have already mentioned, while others will appear later in the book. long-run δ long run = β 1+β 2 1 α 0 notes: (1) As explained in the text, x t+j / h =1, j =0, 1, 2,... (2) x t / h =1, x t+j / h =0,j =1, 2, 3,... If y and x are in logs, the multipliers are in percent

8 Table 4: A model typology. 6 Extensions and examples Type Equation Restrictions ADL y t = β 0 + β 1 x t + β 2 x t 1 + αy t 1 + ε t. None In this subsection we briefly point to several important extensions of the ADL model. Second, to help solidify the understanding of the ADL framework, we provide additional economic examples. Static y t = β 0 + β 1 x t + ε t. β 2 = α =0 DL y t = β 0 + β 1 x t + β 2 x t 1 + ε t α =0 Differenced data 1 y t = β 0 + β 1 x t + ε t β 2 = β 1, α =1 ECM y t = β 0 + β 1 x t +(β 1 + β 2 )x t 1 None +(α 1)y t 1 + ε t Homogenous ECM y t = β 0 + β 1 x t β 1 + β 2 = (α 1) +(α 1)(y t 1 x t 1 )+ε t 1 is the difference operator, defined as z t z t z t Extensions The most important extensions of (8) are: 1. Several explanatory variables 2. Longer lags 3. Systems of ADL equations A few more examples Two exogenous variables, x 1,t and x 2,t. The extension of (8) to this case is y t = β 0 + β 11 x 1,t + β 21 x 1,t 1 (14) + β 12 x 2,t + β 22 x 2,t 1 + αy t 1 + ε t, where β ik is the coefficient of the i 0 th lag of the explanatory variable k. Everything goes through as before, but two different sets of multipliers, with respect to changes in x 1 and x 2. The dynamic consumption function (again) This of course has been the prime example so far in section 3. We have considered the log-linear specification in detail. Of course exactly the same analysis applies to a linear functional form of the consumption function, only that the multipliers will be in units of million kroner (at fixed prices) rather than percentages. In section 7 the linear consumption function is combined with the general budget equation to form a dynamic system. In modern econometric work on the consumption function, more variables are usually included than just income. Hence, there is usually more multipliers to consider than just with respect to INC t. The most commonly found additional explanatory variables are wealth, the real interest rate and indicators of demographic developments

9 The Phillips curve In Chapter 2 of IDM, and several times later in the course, we will consider the Phillips curve: πt+1 e = τπ t 1 is just one out of many hypotheses of expectations formation. Alternative specifications give rise to different dynamic models of the rate of inflation. Consider an explicit inflation target, π. In this case,we may set π t = β 0 + β 11 u t + β 12 u t 1 + β 21 πt+1 e + ε t. (15) π t istherateofinflation, hence π t =ln(p t /P t 1 )wherep t is an index of the price level of the economy. u t is the rate of unemployment, or its log. Finally, πt+1 e denotes the expected rate of inflation one period ahead, so (15) is formally an ADL with 2 explanatory variables. Moreover, if πt+1 e = τπ t 1 (16) (15) can be reduced to the single variable ADL equation (8), π t = β 0 + β 11 u t + β 12 u t 1 + απ t 1 + ε t. (17) with α = β 21 τ. 33 πt+1 e = π (18) As an exercise, convince yourself that equations (15) and (18) imply an equation for inflation which is a distributed lag model (DL model). More generally, firms and households take into consideration the possibility that future inflation is not exactly on target. Hence they may adopt a more robust forecasting rule, for example π e t+1 =(1 τ) π + τπ t 1, 0 <τ 1. (19) In this case, the derived dynamic equation for inflation again takes the form of an ADL model. 34