Stochastic Neoclassical Growth Model

Transcription

1 Stochatic Neoclaical Growth Model Michael Bar May 22, 28 Content Introduction 2 2 Stochatic NGM 2 3 Productivity Proce 4 3. Mean Variance Covariance and Correlation Prediction Summary of AR() propertie Etimation Solving the Model 8 5 Evaluating the Model Performance 8 5. Fact from the data Model economy Running the experiment Concluion 3 7 Appendix A: Analytical Solution 3 8 Appendix B: Two-Period model, State Space Approach 4 9 Appendix C: Equity Premium Puzzle 6 San Francico State Univerity, department of economic.

2 Introduction One of the meage from the Solow model and the Neoclaical Growth Model i that without growth in productivity, it i impoible to achieve utained growth in tandard of living. Both model converge to teady tate, when productivity i contant. In thee note we examine the importance of productivity for buine cycle. In particular, we ak how much of the oberved buine cycle can be accounted for by random hock to productivity. We create a computation experiment in which a tochatic verion on the NGM erve a laboratory for imulated buine cycle. The model i almot identical to the determinitic NGM, but we introduce tochatic (random) productivity. We then can compare the oberved buine cycle fact in the data with the imulated buine cycle in the model. The model preented in thee note i the main workhore for the tudy of buine cycle. Matlab code for olving and imulating thi model are available on the coure web page. 2 Stochatic NGM Jut a with the determinitic NGM, we can prove that tochatic NGM i Pareto e cient. Therefore, the equilibrium can be found by olving the correponding ocial planner problem. We aume that the ocial planner maximize the expected utility of the repreentative houehold, ubject to the feaibility contraint. Thu, the ocial planner problem i: max E fc t;h t;k t+ g t= :t: X t= t u (c t ; h t ) () [Feaibility] : c t + k t+ = A t kt ht + ( ) k t 8t [Productivity] : A t = A ( + A ) t e zt, z t = z t + " t and " t i:i:d: N ; 2 " k > given Thu, the productivity conit of a determinitic part A ( + A ) t, repreenting the growth trend, and tochatic part, e zt, which caue uctuation around the trend. The variable z t i the random productivity parameter, and " t i called white noie proce (alo known a innovation proce). The notation E i conditional expectation, given the information available at time. In general, E t i the conditional expectation given the information available at time t. The information available at time t i k t and A t, and the information available at time i k and A. The Lagrange function i ( X X ) L = E t u (c t ; h t ) t ct + k t+ A t kt h t ( ) k t t= t= We could write the conditional expectation in the uual way, E (X t j t ), where t i the information available at time t. 2

3 The rt order condition for h t ; c t and k t+ are, 8t = ; ; ::: [c t ] : t u (c t ; h t ) = t (2) [h t ] : t u 2 (c t ; h t ) = t ( ) A t kt ht (3) [k t+ ] : t = E t t+ At+ kt+ h t+ + (4) Notice the expectation E t in the Euler equation. Since the optimal path fc t ; h t ; k t+ g t= ha to olve () tarting at any time t, the information on which we condition i that of time t (i.e. k t and A t ). Dividing (3) by (2) give the uual condition for optimal allocation of time (labor upply): u 2 (c t ; h t ) u (c t ; h t ) = ( ) A tkt ht The left hand ide i the marginal rate of ubtitution between leiure and conumption, and the right hand ide i the real wage. Notice that (2) for period t+ i t+ u (c t+ ; h t+ ) = t+. Uing (2) and the ame equation for period t + together with (4), give t = E t t+ At+ kt+ h t+ + t u (c t ; h t ) = E t t+ u (c t+ ; h t+ ) A t+ kt+ h t+ + u (c t ; h t ) = E t u (c t+ ; h t+ ) A t+ kt+ h t+ + The lat condition i called the tochatic Euler equation. The left hand ide how the pain from giving up one unit of conumption and inveting it, while the right hand ide i the expected gain in the next period from extra unit of capital. The neceary condition for optimal fc t ; h t ; k t+ g t= are: u 2 (c t ; h t ) u (c t ; h t ) = ( ) A t k t h t (5) u (c t ; h t ) = E t u (c t+ ; h t+ ) A t+ kt+ h t+ + (6) c t + k t+ = A t kt h t + ( ) k t (7) where A t = A ( + A ) t e zt, z t = z t + " t and " t i:i:d: N ; 2 " In thee note we will aume that A =, ince our goal i to tudy buine cycle (i.e. deviation from trend), and we remove the trend from the data, a well a from the model. At thi point you may be wondering how we ended up with E t when the original problem had E. A very ueful exercie i to tart by deriving the rt order condition for h ; c and k, given the information known at time t =. Uing the above Lagrange, we get the neceary condition fc ; h ; k g: [c ] : u (c ; h ) = [h ] : u 2 (c ; h ) = ( ) A kh [k ] : = E A k h + 3

4 Now that we know k and A and uppoe that we olve the problem in period t =. The lagrange i ( X X ) L =E t u (c t ; h t ) t ct + k t+ A t kt ht ( ) k t t= t= The neceary condition for fc ; h ; k 2 g are: [c ] : u (c ; h ) = [h ] : u 2 (c ; h ) = ( ) A kh [k 2 ] : = E 2 A2 k2 h2 + Similarly, when we olve for fc 2 ; h 2 ; k 3 g, we condition on the information known at t = 2, i.e. we know k 2 and A 2. The lagrange at t = 2 i ( X X ) L =E 2 t u (c t ; h t ) t ct + k t+ A t kt ht ( ) k t t=2 t=2 The neceary condition for fc 2 ; h 2 ; k 3 g are: [c 2 ] : 2 u (c 2 ; h 2 ) = 2 [h 2 ] : 2 u 2 (c 2 ; h 2 ) = 2 ( ) A 2 k2h 2 [k 3 ] : 2 = E 2 3 A3 k3 h 3 + The above tep illutrate that deciion made at time t utilize the information available at that time, and therefore the expectation i taken baed on information available at time t, i.e. E t. 3 Productivity Proce The tochatic productivity i the only ource of uncertainty in thi economy and i the engine of the Real Buine Cycle doctrine (RBC). It i therefore important to dicu in detail the propertie of the probabilitic model that we adopt for modelling the proce of productivity - A t. We aume that Firt, oberve that A t = A ( + A ) t e zt where z t = z t + " t and " t i:i:d: N ; 2 " log (A t ) = log (A ) + t log ( + A ) + z t Thu, we aume that A t uctuate around a contant growth trend with average growth rate of A. Therefore, the log(a t ) ha a linear growth component log (A ) + t log ( + A ) and a cyclical component z t. The model we will analyze in the quantitative ection doe not have the growth component, i.e. we et A =, ince our goal i to generate cyclical 4

5 data (deviation from trend). If we aumed A >, we would have to rewrite the neceary condition in term of detrended variable. Therefore, the productivity in our model i jut A t = A e zt. The parameter A a ect the level of imulated time erie in the model, but not the deviation from the teady tate. Therefore, we typically et A = for implicity. We have already practiced how to decompoe the log of a time erie into a linear trend and the cyclical part, o it hould be clear by now how one can obtain data on z t. Next, we want to dicu the pecial tructure of z t = z t +" t. Thi tochatic proce i called AR(), i.e. rt-order autoregreive proce. It i autoregreive becaue it look like a regreion of z t on itelf, with one lag. An AR(2) proce would be z t = z t + 2 z t 2 +" t, etc. For now, the normality of " t i not important. We will need ome ditributional aumption later, when we imulate the proce of z t. What we focu on i the aumption that " t i i:i:d: (identically and independently ditributed), that E (" t ) = and that V ar (" t ) = 2 ". The lat aumption tate that the variance of " t i contant, and doe not change with time. From thee aumption, we can derive certain propertie of z t. 3. Mean Firt, we would like to nd the mean and variance of z t. For thi purpoe, we recurively ubtitute the expreion of z t into z t : z t = (z t 2 + " t ) + " t = 2 z t 2 + " t + " t Next, ubtitute z t 2 z t = 2 (z t 3 + " t 2 ) + " t + " t = 3 z t " t 2 + " t + " t Thi recurive ubtitution become in the limit z t = lim k! k z t k + X k " t k= k We will alway aume that jj <, o the rt term vanihe. Thi aumption i called tationarity condition. Thu, z t = " t + " t + 2 " t 2 + ::: Oberve that z t i a weighted average of all the pat hock " t ; " t ; " t 2 ; ::: Thi i called the moving average repreentation of z t. Since E (" t ) = 8t, we have E (z t ) = E (" t ) + E (" t ) + 2 E (" t 2 ) + ::: = 5

6 3.2 Variance V ar (z t ) = V ar (" t ) + 2 V ar (" t ) + 4 V ar (" t 2 ) + ::: = 2 " " " + ::: X = 2 " 2 t 2 = " 2 t= Notice that we ued the aumption that the " are independent, which implie that the variance of the um i the um of the variance. 3.3 Covariance and Correlation Cov (z t ; z t ) = Cov (z t + " t ; z t ) = Cov (z t ; z t ) + Cov (" t ; z t ) = V ar (z t ) + The lat term i zero becaue z t depend on all the previou hock " t ; " t 2 ; :::, which are not correlated with the current hock " t. Similarly, Cov (z t ; z t 2 ) = Cov (z t + " t ; z t 2 ) = Cov (z t ; z t 2 ) = 2 V ar (z t ) Cov (z t ; z t k ) = k V ar (z t ) Corr (z t ; z t k ) = Cov (z t ; z t k ) p V ar (zt ) p V ar (z t k ) = k The covariance tell u that the proce i peritent, i.e. adjacent hock are correlated, but thi correlation diminihe when the two hock are far apart. For example, if = :95, then Corr (z t ; z t ) = :95 and Corr (z t ; z t 25 ) = :95 25 = : Prediction Suppoe that we know the value of the hock at time t and we want to make a prediction about the next hock. Thu, we want to nd the conditional expectation of z t+ given z t. E (z t+ jz t ) = E (z t + " t+ jz t ) = E (z t jz t ) + E (" t+ jz t ) = z t + The lat term i zero becaue " are independent of each other, o " t+ i independent of all the pat ". Conditional expectation, i equal to unconditional expectation, when we 6

7 condition on independent variable. Thu, E (" t+ jz t ) = E (" t+ ) =. Suppoe that we want to predict two period ahead: Similarly, E (z t+2 jz t ) = E (z t+ + " t+2 jz t ) = E (z t+ jz t ) = 2 z t E (z t+k jz t ) = k z t For example, uppoe that = :9 and the hock today i z t = 3. What i your prediction of z t+ and z t+? Anwer: E (z t+ jz t = 3) = :9 3 = 2:7 E (z t+ jz t = 3) = :9 3 = :46 Thu, we predict that the next period hock will be cloe to today, but after period there i a decay due to < <. 3.5 Summary of AR() propertie Suppoe that Then 3.6 Etimation z t = z t + " t where " t i:i:d: with E (" t ) =, V ar (" t ) = 2 " [Mean] : E (z t ) = [Variance] : V ar (z t ) = 2 " 2 [Autocovariance] : Cov (z t ; z t k ) = k 2 " 2 [Autocorrelation] : Corr (z t ; z t k ) = k [Prediction] : E (z t+k jz t ) = k z t The parameter of thi probabilitic model are and ". The implet way to etimate i running the regreion z t = z t + " t ) ^ and ^ " can be found from the reidual of the above regreion, i.e. v u ^ " = t nx (z t ^z t ) 2 n t= 7

8 Alternatively, we can etimate V ar d(z t ) = n nx t= And then, ue V ar (z t ) = 2 " 2 to etimate " (z t E (z t )) 2 = n nx t= z 2 t d V ar (z t ) = 2 " ^ 2 ) ^ 2 " = d V ar (z t ) ^ 2 4 Solving the Model The code poted on the coure web page ue the method called "extenion of determinitic path". Thi i eay to implement method, but not very e cient in term of computational time. Suppoe that at time t = we oberve a value z, and uppoe that there are no more hock in the future: " t = 8t >. In thi cae the future value of z t are a follow: z = z, z 2 = 2 z,...,z t = t z,... Thu, z t! a t!, and e zt! a t!, and eventually the model will converge to the teady tate, a gure how. Thee graph are called impule repone function, ince they how the repone of endogenou variable to a one-time exogenou hock to technology. Solving the determinitic model with no more hock after the very rt period, will give u the value of (c ; h ; k ) that are approximately correct. Thi i becaue the houehold chooe (c ; h ; k ) at time, and the optimal choice of thee variable doe not depend on future hock. Thu, we can olve the determinitic model given fz t g T t= = ft z g T t=, and T i ome large nal period when the ytem i cloe to teady tate. Notice that by olving the determinitic model with f t z g ue an approximation of the Euler equation E t (f (z t+ )) f (E t (z t+ )), for ome non-linear function f. By Jenen inequality, we know that the above i only equal for linear function f, and that i the reaon for the ign. Thu, olving for the determinitic path give approximately correct value of initial capital for the next period, k. Now, uppoe that in the next period the economy i hit by another hock, " 6=. Then, z = z + ". If there will not be any hock after t =, the model will converge to a teady tate, and we can olve for the determinitic path that would give an approximately correct olution for k 2. By repeatedly olving for determinitic path from time t onward, we obtain a equence of endogenou variable for the model. 5 Evaluating the Model Performance In order to evaluate the model, we need to tart with peci c quetion, for example "how much of the oberved uctuation in the U.S. economy can be accounted for by hock to productivity?". The rt tep in anwering that quetion i to document ome buine cycle fact from the data. After that, the model i calibrated, and imulated to generate arti cial data. Finally, when we have the fact from the data and the imulated obervation from 8

9 path of c det 5 path of h det 3 path of k det c det.5 h det 5 k.5 det path of A det path of y det path of w det A.5 det. y det. w det path of r det.5 path of x det r det x det Figure : Impule Repone Function the model, we can evaluate the performance of the model. We can ak how well doe the model replicate the fact from the data. The above procedure i called the computational experiment. 5. Fact from the data There are many buine cycle fact that one might want to document. At the mot baic level, economit are intereted in volatility and comovement. Volatility refer to the deviation of key economic variable from ome trend. The next gure how one way that economit ue to decompoe a time erie into trend and cyclical part. The top panel how the natural log of real GDP, real peronal conumption expenditure and real invetment, together with a linear trend. The bottom graph how the deviation of the log(data) from it linear trend. The bottom graph therefore depict the detrended 9

10 log(real I per capita) log deviation from trend log(real C per capita) log deviation from trend log(real GDP per capita) log deviation from trend log(real GDP per capita) and trend. log(real GDP per capita) cyclical I 957 I 967 I 977 I 987 I 997 I 27 I. 947 I 957 I 967 I 977 I 987 I 997 I 27 I.5 log(real C per capita) and trend. log(real C per capita) cyclical I 957 I 967 I 977 I 987 I 997 I 27 I. 947 I 957 I 967 I 977 I 987 I 997 I 27 I 9 log(real I per capita) and trend.4 log(real I per capita) cyclical I 957 I 967 I 977 I 987 I 997 I 27 I I 957 I 967 I 977 I 987 I 997 I 27 I data or the cyclical part of the data. Economit are intereted in meauring the volatility of the cyclical part, a meaured by it tandard deviation. The number on the vertical axi give approximately the percentage deviation of the original variable from the trend. For example, notice that the deviation of invetment from it trend are much larger than the deviation of output from the trend. Thi i re ected in larger tandard deviation of the detrended invetment. We therefore ay that invetment i more volatile than output in the U.S. economy. Comovement decribe how the movement of key variable coincide with the movement of real GDP. If deviation from trend of ome variable are in the ame direction a thoe of real GDP, we ay that the variable i pro-cyclical, and if the deviation are in the oppoite direction of real GDP, the variable i called counter-cyclical. A convenient tatitic for meauring comovement i the correlation coe cient.

11 5.2 Model economy The main workhore for the tudy of buine cycle i the neoclaical growth model. max E :t: X t= t u (c t ; h t ) [Feaibility] : c t + k t+ = e zt kt ht + ( ) k t where z t = z t + " t and " t i:i:d: N ; 2 " We calibrate the model economy to match ome long-term moment in the data. For thi purpoe we need to aume that there are no technology hock, o that the model will converge to a teady tate. Firt we need to chooe the functional form for the utility function: u (c t ; h t ) = log c t + ( ) log ( h t ) The rt order condition become: At teady tate: c t = ( ) kt ht h t c t+ = kt+ ht+ + c t c t + k t+ = kt ht + ( ) k t c h = ( ) y h = h y i k + (8) (9) c + k = y () From the NIPA, we know that = :35. From equation () we have c k + = y k = y k The value of depend on the frequency of the data (annual, quarterly). From equation (9) we nd. = h y i k + = y k + To calibrate we need to ue equation (8), and ome evidence on the fraction of time endowment allocated to work. For example, if the dicretionary time endowment i c k

12 hour per week and the average workweek i 4 hour, then we et h = :4 in equation (8) and olve for. c h = ( ) y h = ( ) h h = + B y c B In addition to the above parameter, we need to calibrate the parameter that govern the TFP, i.e. and ". Firt, we need to obtain data on TFP, uing Y t = A t K t L t ) A t = Y t K t L Thi require data on real output, real tock of phyical capital and ome meaure of labor input - uually index of aggregate hour. Since we aume that A t = A ( + A ) t e zt, to obtain a erie of fz t g one need to take log, and run the regreion log (A t ) = log (A ) + t log ( + A ) + z t or log (A t ) = + t + z t The reidual of thi regreion give u an etimate of fz t g. Finally, to etimate we run the regreion of z t on it lag, i.e. z t = z t + " t Thi give the etimated value of. The etimate of " i obtained imply a the tandard deviation of the reidual from the lat regreion. 5.3 Running the experiment The lat tage in our computational experiment conit of uing the calibrated model to generate arti cial data. Thi can be done with code like RBC.m that upplement thi chapter. We compute the volatility and comovement tatitic for the arti cial data, and preent it ide by ide with their data counterpart. The next table i an example of what the comparion of data with the model might look like. t 2

13 In thi example, the model generate about 75% of the oberved volatility in output. Remember that the only ource of uctuation in the model i productivity hock, while in the data cal and monetary policy hock might contribute to the oberved uctuation. The model alo capture the conumption moothing behavior oberved in the data, in that conumption i le volatile than output and invetment i more volatile. The model generate much greater volatility in invetment than in the data. Moreover, the model generate much maller volatility in the labor market variable than in the data. 6 Concluion We contructed a model economy to imulate buine cycle and compare the imulated reult to the data. The model capture ome feature from the data well and other not o well. The next tep in buine cycle reearch i to try and undertand what feature of the model are reponible for the failure and modify the model accordingly. For example, ome economit introduce more elaborate tructure into the labor market, in the form of ticky wage or earch. One advantage of thee extenion i that the model can generate pattern of unemployment, imilar to the one oberved. Thee and other extenion are beyond the cope of thi coure. 7 Appendix A: Analytical Solution A pecial cae of the tochatic NGM ha analytical olution. Conider a model with inelatic labor upply, log utility, u (c t ) = log (c t ), Cobb-Dougla production function, y t = A t k t, and full depreciation, =. We can how that the optimal policy for invetment i k t+ = y t 3

14 and for conumption c t = ( ) y t. The Euler Equation become: = E t A t+ kt+ c t c t+ ct or = E t A t+ kt+ c t+ Subtitute the propoed policie into the Euler Equation: ( ) At kt = E t A ( ) A t+ kt+ t+ kt+ = E t A tkt k t+ Plug k t+ = A t k t, and implify: = E t A tkt A t kt = E t ) = Thu, an optimal policy i k t+ = A t k t, c t = ( ) A t k t. 8 Appendix B: Two-Period model, State Space Approach Conider a 2-period conumption and aving/invetment problem under uncertainty. The current period tate i known, but uncertainty about the econd period i repreented by di erent poible tate of nature, indexed = ; 2; :::; S. Let be the probability that tate occur. The houehold receive wage w t in the current period, and chooe conumption c t, c t+ at tate = ; 2; :::; S, and aet k t+. The rate of return on aet in the next period r t+ i unknown in period t, and depend in tate of nature. Thu, we denote the rate of return in tate by r t+. We can aume that future wage i alo uncertain, i.e. w t+ i the next period wage in tate (although thi generality doe not a ect our main point). The houehold income in the econd period, in tate, i + r t+ kt+ + w t+. The houehold problem i: max c t;c t+ ;k t+ :t: u (ct ) + u c t+ = [BC t ] : c t + k t+ = w t [BC t+ ] : c t+ = + r t+ kt+ + w t+, = ; 2; :::; S 4

15 Notice that in the econd period, there are S poible budget contraint, and only one of them will actually realize. But at time t, the houehold doe not know which budget contraint will realize. The Lagrange function aociated with the above problem i L = u (c t ) + = Firt order condition: u c t+ t [c t + k t+ w t ] = t+ [c t ] : u (c t ) t = c t+ : u ct+ t+ =, = ; 2; :::; S [k t+ ] : t + t+ + rt+ = = c t+ + rt+ kt+ wt+ Notice that unit of aet k pay return in all tate of nature. Plugging t and t+ from the rt two condition into the Euler Equation, give: t = u (c t ) = t+ + rt+ = = Notice that the lat equation can be written a u (c t ) = E t u u c t+ + r t+ c t+ + r t+ Compare thi tochatic Euler Equation with the one in (6). The above optimal invetment condition can be derived from a lightly modi ed, equivalent Lagrange function: L = u (c t ) + L = = u c t+ u (ct ) + u c t+ = t [c t + k t+ w t ] = t [c t + k t+ w t ] t+ () t+ c t+ + rt+ kt+ wt+ c t+ + rt+ kt+ wt+ where t+ t+= i caled multiplier. Notice that we did not change the original Lagrange function, and that it can be written in expected value form: L = E t u (ct ) + u c t+ t [c t + k t+ w t ] t+ c t+ + rt+ kt+ wt+ The rt order neceary condition are [c t ] : u (c t ) t = c t+ : u c t+ t+ =, = ; 2; :::; S [k t+ ] : t + t+ + rt+ = = 5

16 Plugging t and t+ from the rt two condition into the Euler Equation, give: t = u (c t ) = = = t+ + rt+ u c t+ + r t+ Notice that the lat Euler Equation i identical to (). Thu, we can derive the rt order neceary condition for a dynamic tochatic contrained optimization problem, from the Lagrange in expected value form, with all the contraint brought inide the expectation operator. The only di erence i that the interpretation of t+ i the ame a t+, the e ect of a unit increae in income in tate on the maximized level of expected utility (objective function). The multiplier t+ on the other hand i the value of unit increae in income in tate on utility in that tate. 9 Appendix C: Equity Premium Puzzle Baed on the lat ection, = E t u (c t+ ) u (c t ) ( + r t+) = E t [m t+ ( + r t+ )] where m t+ = u (c t+ ) u (c t) i tochatic dicount factor. The right hand ide can be written a = cov t [m t+ ; + r t+ ] + E t (m t+ )E t ( + r t+ ) For rik-free aet, we have h i = E t m t+ + r f t+ = E t (m t+ ) + r f t+ Combining with riky aet cov t [m t+ ; + r t+ ] + E t (m t+ )E ( + r t+ ) = E t (m t+ ) + r f t+ E t ( + r t+ ) = + r f cov t [m t+ ; + r t+ ] t+ E t (m t+ ) E t (r t+ ) = r f cov t [m t+ ; + r t+ ] t+ E t (m t+ ) Plugging in the tochatic dicount factor m t+ = u (c t+ ), u (c t) h i cov t u (c t+ ) ; ( + r E t (r t+ ) = r f u (c t) t+) t+ h i E t u (c t+ ) E t (r t+ ) r f t+ = u (c t) cov t [u (c t+ ) ; ( + r t+ )] E t (u (c t+ )) 6

17 The term on the right i rik adjutment or rik premium or equity premium. Thu, if an aet return i uncorrelated with conumption, it expected return i equal to the rik free return, and there i no premium. Since u (c) i diminihing, aet return that are poitively correlated with conumption are negatively correlated with marginal utility. Thu, aet with return that are poitively correlated with conumption, hould have cov t (u (c t+ ) ( + r it+ )) <, mut promie higher expected return (becaue thee aet make conumption more volatile, or increae rik). On the other hand, aet that are negatively correlated with conumption, and therefore cov t (u (c t+ ) ( + r t+ )) >, can o er expected return that are lower than the rik free return (becaue they help mooth conumption, or reduce rik - erve a inurance). The equity premium puzzle refer to the phenomenon that oberved return on tock over the pat century are much higher than return on government bond. It i a term coined by Rajnih Mehra and Edward C. Precott in 985, and howed that either a large rik averion coe cient or counterfactually large conumption variability wa required to explain the mean and variance of aet return. Economit expect arbitrage opportunitie would reduce the di erence in return on thee two invetment opportunitie to re ect the rik premium invetor demand to invet in relatively more riky tock. Reference [] Mehra, Rajnih. "The equity premium: why i it a puzzle?." Financial Analyt Journal 59, no. (23):