Traditional treasury bills and TIPS

TIPS----Treasury Inflation-Protected Securities.

As its name implies, TIPS' major property is that the principal is adjusted to nation's price level, usually measure as CPI. For example, suppose the coupon rates of a ten-year TIPS and a ten-year traditional treasury bill are both 3 percent. If investors buy them at par and hold to maturity, and average inflation is 2.5 for the next 10 years, then the real return of the traditional treasury bill is only 0.5 percent while that of the TIPS is still 3.5 percent. (WHY? Because the principal coupon rate is automatically adjusted to be 2.5+3=5.5 percent) So TIPS keeps the purchasing power.

Ideally, the return yield between TIPS and traditional treasury bill is the average market participants' inflation expectation, which can give us valuable information in inflation forecasting. In real life, however, some other factors also play roles in this yield spread.

(1) Inflation risk:
TIPS is inflation-risk free, while traditional treasury bill is subject to inflation effect, so investors bear more risk buying the latter bond. Based on economic theory, investors should be compensated by some amount to buy the bond, and we call this amount of pecuniary compensation as Inflation Risk Premium.

(2) Liquidity risk:
Based on finance theory, value of a certain type of asset is related to its liquidity. If the asset takes significant time and effort to sell at fair price, then investors might have to lower price to sell it when time constraint is tight. Traditional treasury bills are supposed to be the most liquid assets in financial market, and thus investors of these bonds don't have such worries. TIPS are relatively more illiquid and thus liquidity premium might exist to attract investors to buy TIPS. (TIPS have higher liquidity risk)

To summarize, the yield spread can be expressed as:

y^(n) - y^(r) = inflation expectation + inflation risk premium - liquidity risk premium,

where  y^(n) denotes nominal return of traditional treasury bills, and y^(r) denotes real return of TIPS.

From this equation it is clear that the spread conveys accurate info about inflation expectation only when inflation risk premium is of the same size as liquidity risk premium. However, if both premium are significant smaller than inflation expectation, then the spread still has a good approximation of inflation expectation. Furthermore, if premiums are roughly constant over time, then we can know change of inflation expectation by tracking change of spread rate.

Paper review of Stock and Watson's Forecasting Inflation

The paper I review is John Stock and Mark Watson's \textit{Forecasting Inflation}, which was published in 1999. The main focus of the paper is on Phillips curve's forecast ability, and authors aimed to answer the four following questions. Has the curve been stable over time? Has the unemployment rate-based Phillips curve done a good job predicting U.S. inflation at 12-month horizon? Is it possible that the curve can be improved by incorporating some other real economic activity variables? If not, does there exist any other model that we can resort to? 

First introduced by British economist William Phillips in 1958, the Phillips curve shows that rising price level helps decreasing unemployment rate. However, this inverse relationship doesn't hold in the long run because market participants tend to adjust their inflation expectation and thus as time goes on price level increases while unemployment rate returns back to its natural level. (Friedman, 1968) So in terms of forecasting inflation, researchers typically limit prediction horizons to 12-month so that the inverse relationship exists.

In their paper, Stock and Watson (abbreviated as S\&W in the rest of the paper) broadly interpreted Phillips curve as short-run relationship between future inflation and current real market activity. In other words, they aggregate variables in addition to unemployment rate in regressions and examined if they help improve forecast performance of the traditional Phillips curve. To discover existence of better alternative prediction tools, S\&W also considered models either backed by economic theories or more sophisticated in model construction. S\&W used U.S. monthly data from 1959 January to 1997 September on inflation as measured by CPI and Personal Consumption Expenditure (PCE) deflator. The data source is DRI-McGraw Hill Basic Economics Database.

The base model S\&W used is as follows: $$\pi^{h}_{t+h}-\pi_{t}=\phi+\beta(L)u_{t}+\gamma(L)\Delta\pi_{t}+e_{t+h}$$ where 

\begin{enumerate}

\item $\pi^{h}_{t+h}=\frac{1200}{h}\ln (\frac{P_{t+h}}{P_{t}})$: h-period ahead inflation in the price level $P_{t+h}$, reported at an annual rate

\item $\pi_{t}=1200 \ln(\frac{P_{t}}{P_{t-1}})$: monthly inflation at annual rate

\item $u_{t}$: unemployment rate

\item $\Delta\pi_{t}$: first order difference of inflation

\item $\beta(L)$, $\gamma(L)$: polynomials in the lag operator L 

\end{enumerate}

There are two assumptions in the model: inflation data has a unit root and the natural rate of unemployment (NAIRU) is constant.\footnote{Indeed, the natural rate $\bar{u}$ is captured by the constant term $\phi$}. These two assumptions are common in related literature but if future researches find strong evidence against them, then the validity of S\&W's tests are questionable.

S\&W first tested whether coefficients in the model have been stable over the sample period. They used Quandt likelihood ratio (QLR) test to discover unknown breakdate. The null hypothesis of QLR(all) is that all coefficients are jointly stable; QLR($\phi,\beta$) tests whether $\phi$ and $\beta(L)$ are jointly stable, assuming constancy of $\gamma(L)$, and QLR($\gamma$) is just the opposite. Table.1 shows their testing result. QLR($\gamma$)s have small p-values, implying that this variable is unstable, and the structural break is 1983. However, QLR test may not be the best testing method, and there could exist more than one structural break. Even though they found model was unstable, S\&W ignored it due to quantitative insignificance of $\gamma(L)$ and regarded it as the benchmark model. 

S\&W broadly interpreted Phillips curve as short run relationship between future inflation and current real market activities. The generalized model is as follows: $$\pi^{h}_{t+h}-\pi_{t}=\phi+\beta(L)x_{t}+\gamma(L)\Delta\pi_{t}+e_{t+h},$$ where $x_{t}$ denotes real market variables other than unemployment rate and is either processed or assumed to be stationary. Their detrending method is one-sided Hodrick-Prescott (HP) filter. 

 They first used recursive method to estimate in-sample model coefficients and then conducted pseudo out-of-sample forecast. To compare these models' forecast performance, authors used relative mean square error (Rel.MSE) with respect to benchmark Phillips curve and forecast combination regression method. Namely, they regressed the following model: $$\pi^{h}_{t+h}-\pi_{t}=\lambda f^{X}_{t}+(1-\lambda)f^{U}_{t}+\epsilon_{t+h},$$ where 

\begin{enumerate}

\item $f^{X}_{t}$ denotes forecasts based on candidate variable X, made at time t

\item $f^{U}_{t}$ denotes forecasts based on unemployment rate, made at time t

\item $\lambda$ shows how much forecast based on x helps improve forecast ability of benchmark unemployment forecast. 

\end{enumerate}

If Rel.MSE is statistically less than 1, then it means its corresponding forecast is better than the benchmark. If $\lambda$ is statistically positive, then the corresponding variable adds forecast accuracy to the benchmark based on unemployment rate.

Table.2 and Table.3 tabulated their results. We can observe that in general PCE inflation forecasts are most accurate than CPI inflation ones. Forecast results in the second-half period (1984-1996) are more accurate than the first half. What's more, over years the variables \textit{capacity utilizations} (i.e.IPXMCA) and \textit{manufacturing and trades sales} (i.e.MSMTQ) both have small Rel.MSEs and positive $\lambda$s. So these two variables  help improve forecast that is based on unemployment rate. 

To discover potential existence of better models, S\&W also considered models backed by term structure expectations, nominal money supply theory, and multivariate models incorporated with different kinds of variables.\footnote{S\&W used the same methods of coefficient estimation and forecast performance comparison. Since the result tables are too large, I choose not to include them in the paper.} They found that: (1) bivariate models based on term structure of interest rates or nominal money supply produced worse forecast results than Phillips curve; (2) bivariate models incorporating exchange rate or different price indexes didn't outperform Phillips curve consistently; (3) multivariate regressions showed that incorporation of other real activity variables didn't significantly improve the forecasting results. 

Based on their findings, S\&W concluded that Phillips curve produced the most reliable short-run inflation forecast among all models they considered and certain aggregate activities variables, namely capacity utilization and manufacturing and trade sales help improve the curve's forecast ability. Their conclusions echoed with some previous researches. For example, Alan Blinder (2001), the former vice chairman of the Board of Governors of the Federal Reserve System, claimed that the curve has done amazingly well over years and it deserves a prominent place in models used for policy-making purposes.  

However, there are certain drawbacks in this paper. First of all, since the authors used a large number of forecasts, overfitting bias is inevitable. In addition, they assumed that inflation is I(1), which is controversial. If future researches evidently prove that inflation is integrated of other orders, then this paper's findings might not be valid. Furthermore, S\&W only considered linear models for the sake of simplicity, but as they admitted, the relationship between inflation and real economy variables can be much more complicated. Indeed, Ang, Bekaert and Wei (2006) argued that this paper failed to consider non-linear models like no-arbitrage term structure models or various survey data and thus their findings are incomplete and unconvincing. The harshest critique, however, comes from Atkeson and Ohanian (2001). They used the following random walk model as the benchmark  $$\pi^{12}_{t+12}=\pi^{12}_{t}+v^{12}_{t+12},$$ where $\pi^{12}_{t+12}$ is the forecast of the 12-month rate of inflation and $v^{12}_{t+12}$ is white noise, and used the same forecasting comparison methods as S\&W did. Surprisingly, they found that from 1985 to 2000, none of S\&W's models performed better than this naive random-walk model. This suggests that the best indicator of future inflation is current price level, which, if true, makes researchers' models based on sophisticated theories pretty futile. However, as S\&W commented in their later paper (2008), the robustness of Atkeson-Ohanian model depends delicately on sample periods and forecast horizon. 

Forecasting inflation is hard and Phillips curve is merely one candidate model. Recently some researchers (Spen and Corning (2001), Calstrom and Fuerst (2004)) start to use Treasury Inflation-Protected Securities (TIPS) for price level prediction. As its name implies, TIPS are the inflation-indexed bonds whose principal is adjusted to the CPI. If CPI changes, the principal adjusts correspondingly to maintain the same purchasing power. By studying behavior of TIPS, we can gain information on investors' expectation of inflation and do forecast based on that.

References

\item Friedman, Milton, "The Role of Monetary Policy", American Economic Review, March 1968, pp 1-17

\item Blinder, Alan, "Is There A Core of Practical Macroeconomics That We Should Believe?", American Economic Review, May 1997, pp 240-243.

\item Ang, Andrew, Green Bekaert and Min Wei, "Do Macro Variables, Asset Markets, or Surveys Forecast Inflation Better?" Journal of Monetary Economics, May 2007, pp. 1163-1212

\item Atkeson, Andrew and Lee Ohanian, "Are Phillips Curve Useful for Forecasting Inflation?", Quarterly Review, The Federal Reserve Bank of Minneapolis, 2001

\item Stock, John and Mark Watson, "Phillips Curves Inflation Forecasts", NBER Working Paper 14322, September 2008

\item Spen, Pu and Jonathan Corning, "Can TIPS Help Identify Long-term Inflation Expectations?", Federal Reserve Bank of Kansas, Economic Review, Fourth Quarter, 2001

\item Carlstrom, Charles and Timothy Fuerst, "Expected Inflation and TIPS", Federal Reserve Bank of Cleveland, November 2004

Tables:

Fibonacci Sequence (LaTex file)

In mathematics, Fibonacci sequence is a sequence of positive integers 

$$a_{1}, a_{2}, a_{3},\cdots a_{n} $$

which satisfies the following conditions

\begin{enumerate}

\item $a_{1}=1$,

\item $a_{1}=1$,

\item $a_{n}=a_{n-1}+a_{n-2}, \forall n \geqslant 2$

\end{enumerate}

This is a recurrence relation and we need to find out a general formula to denote every single member of the sequence. 

There are many ways to solve for this question. For example, a combinatorial expert may define a generating function $F(x)=\sum^{\infty}_{n=1} a_{n}x^{n}$ where $a_{i}$ is a Fibonacci number.\footnote{For a detailed proof of this method, refer to Richard Brualdi's \textit{Introductory Combinatorics}, 5th edition.}  By solve for $F(x)$ explicitly, we can write out the general formula of the sequence: $${a_{n}=(\frac{1}{\sqrt{5}})(\frac{1-\sqrt{5}}{2})^{n}-\frac{1}{\sqrt{5}} (\frac{1+\sqrt{5}}{2})^{n}}$$.

Another widely recognized proof uses matrix. By noticing the recurrence relation is a linear difference equation, we can set up a two-dimensional system

\[

                \left(   \begin{array}{r} a_{n+2} \\ a_{n+1}  \end{array}\right)= \left(   \begin{array}{rr} 1 & 1 \\ 1 & 0  \end{array}\right)\times \left(   \begin{array}{r} a_{n+1} \\ a_{n}  \end{array}\right).

\]

Using eigenvalues and respective eigenvectors of this system, we can also get the same general formula of the sequence. \footnote{For a more detailed proof, refer to Henry Gould (1981)}

In this post, I will use another method to prove the sequence's general formula. The methodis conceptually straightforward and requires no further knowledge than basic geometric sequence properties and undetermined coefficient calculation. In specific, I will construct a couple of geometric sequences from the relation $a_{n}=a_{n-1}+a_{n-2}$, calculate their general formulas, and combine them to get the final result. 

First of all, we know that for a geometric sequence {$b_{n}$}, if $b_{1}=k$ and its recurrence relation is $b_{n}=mb_{n-1}$, ($k \in \mathbb{R}, m \in \mathbb{R}$), then we know $b_{n}=k m^{n-1}$.
Given the recurrence relation $a_{n}=a_{n-1}+a_{n-2}, n \geqslant 3$, it is possible to rewrite this equation and construct a geometric sequence, and by writing out its general formula, we are one step closer to calculating the general form of Fibonacci numbers. 
Rewrite   $a_{n}=a_{n-1}+a_{n-2}$ to be $a_{n}+y(a_{n-1})=x(a_{n-1}+ya_{n-2})$, and this implies that if we can find real numbers $x$ and $y$ to make the two equations identical, then the sequence ${a_{n}+ya_{n-1}}$ is a geometric sequence with common ratio $x$. 

After some calculations, we can get $a_{n}+y(a_{n-1})=x(a_{n-1}+ya_{n-2})$ is $a_{n}=(x-y)a_{n-1}+xya_{n}$, and thus it is obvious that \begin{subequations}
\begin{align}
        x-y &= 1,\\
        xy &= 1,
\end{align}
\end{subequations}
Thus, $x=\frac{-1 \pm \sqrt{5}}{2}$ and $y=\frac{1 \pm \sqrt{5}}{2}$. We choose $x$ to be $\frac{1+\sqrt{5}}{2}$, and $y$ is $\frac{-1+\sqrt{5}}{2}$ correspondingly \footnote{Choosing the other bundle yields the same final result.}.
Now that $a_{n}+\frac{-1+\sqrt{5}}{2}(a_{n-1})=\frac{1+\sqrt{5}}{2}(a_{n-1}+\frac{-1+\sqrt{5}}{2}(a_{n-2}))$, we can see that $a_{n+1}+\frac{-1+\sqrt{5}}{2}(a_{n})$ is a geometric sequence whose initial term is $1+\frac{-1+\sqrt{5}}{2}=\frac{1+\sqrt{5}}{2}$ and common ratio is $\frac{1+\sqrt{5}}{2}$. So  $$a_{n+1}+\frac{-1+\sqrt{5}}{2}(a_{n})=(\frac{1+\sqrt{5}}{2})^n.$$
Seeing that the strategy of constructing geometric sequence works in the first stage, we might be attempted to do it again to the above recurrence relation. However, it is important to note that the right hand side term is no longer a constant; instead, it is also related to $n$. Thus, we need to add rewrite the relation slightly differently. 
Let $a_{n+1}+\lambda (\frac{1+\sqrt{5}}{2})^{n+1}=\frac{1-\sqrt{5}}{2}(a_{n}+\lambda (\frac{1+\sqrt{5}}{2})^{n})$, and compare it to the original equation $a_{n+1}+\frac{-1+\sqrt{5}}{2}(a_{n})=(\frac{1+\sqrt{5}}{2})^n$, we know to equalize the two equations, it must be the case that $$\lambda (\frac{1-\sqrt{5}}{2}) (\frac{1+\sqrt{5}}{2})^{n}-\lambda (\frac{1+\sqrt{5}}{2})^{n+1}=(\frac{1+\sqrt{5}}{2})^{n}$$
After some computation work, we can get that $\lambda=-\frac{1}{\sqrt{5}}$. Plug it back in the recurrence relation, we get $$a_{n+1}-\frac{1}{\sqrt{5}} (\frac{1+\sqrt{5}}{2})^{n+1}=\frac{1-\sqrt{5}}{2}(a_{n}-\frac{1}{\sqrt{5}} (\frac{1+\sqrt{5}}{2})^{n})$$

Although it looks a little bit ugly, actually the sequence ${a_{n}-\frac{1}{\sqrt{5}} (\frac{1+\sqrt{5}}{2})^{n}}$ is a geometric sequence with initial term $\frac{5-\sqrt{5}}{10}$ and common ratio  $\frac{1-\sqrt{5}}{2}$. So $${a_{n}-\frac{1}{\sqrt{5}} (\frac{1+\sqrt{5}}{2})^{n}}=(\frac{5-\sqrt{5}}{10})(\frac{1-\sqrt{5}}{2})^{n-1}$$. 

Note that $\frac{5-\sqrt{5}}{10}=(\frac{1}{\sqrt{5}})(\frac{1-\sqrt{5}}{2}).$ If we rewrite the equation, then $${a_{n}=(\frac{1}{\sqrt{5}})(\frac{1-\sqrt{5}}{2})^{n}-\frac{1}{\sqrt{5}} (\frac{1+\sqrt{5}}{2})^{n}}$$
and this is the exact general formula of Fibonacci sequence. 

My method stems from realization that potential geometric sequences might be derived from the given recurrence relation, and since we are more familiar with geometric sequence's properties, we can use it to tackle more complicated problems. The method is conceptually naive and requires the original relations to be simple. Otherwise, we may resort to other techniques like generating functions.

REFERENCE
1. Brualdi, Richard, \textit{Introductory Combinatorics, 5th edition}
2. Gould, Henry \textit{A History of the Fibonacci Q-Matrix and a Higher-dimensional Problem}, Fibonacci Quart., 19(1981), 250-257

Fermat's Last Theorem

The Fermat's Last Theorem states that for $a, b, c \in \mathbb{Z^+}$ and $n \in \mathbb{N}, n > 2$, the equation $ a^n + b^n = c^n$ doesn't hold. 

Despite its apparent simplicity in terms of hypothesis and conclusion, this theorem is actually very involved and remained a mere conjecture until Andrew Wiles formally proved it in 1995, more than 300 years after Fermat first wrote it in the margin of Diophantus's \textit{Arithmetica}. 

The page of \textit{Arithmetica} that motivated the question was about Pythagorean theorem, which states that in any right triangle, the square of hypotenuse is equal to the sum of square of two other sides. In other words, $\exists a,b,c \in \mathbb{Z^+}$ such that $a^2+b^2=c^2$.   

Having tested various solutions to the Pythagorean theorem, Fermat went a step further to check whether the theorem holds for $n=3$. Surprisingly, Fermat argued that such system is insoluble and more generally, there is no positive integer solutions to any n that is larger than $2$. He commented that he came up with a remarkable proof to solve for all cases but the margin is too narrow for him to write down.\cite{citationkey1} As a result, we are only able to read Fermat's proof for a special case where $n=4$, while the general theorem remained unsolved for the next three centuries. 

Mathematicians at first tried to find existence of counter-examples but failed, which incentivized more people to believe that Fermat's conjecture is true. Later, mathematicians managed to prove the conjecture when $n=3, 5$ and $7$, but no one was able to come up with a general proof for all equations. 

The longer one conjecture remained mysterious, the more attractive it seemed to people who inspired to solve it. One story says that in nineteenth century a German physician and mathematician Paul Wolfskehl bequeathed 100,000 marks (quite a big amount of money at that time) to the first one who could prove it. Such lucrative prize motivated more people, especially amateurs to hand in their proofs, but unfortunately none of their attempt was successful. Due to advent of modern computers after WWII, mathematicians were able to conclude that Fermat's conjecture holds for finitely large $n$, but still, nobody was able to give a complete proof. The conjecture also became notorious for its difficulty and before Wiles' proof, it was described as "most difficult mathematical problems" in the \textit{Guinness Book of World Records}. \cite{2}

In the twentieth century, Shimura and Taniyama, two Japanese mathematicians specialized in elliptic curves, conjectured that each elliptic curve can be matched to a modular. It seemed at first Shimura-Taniyama conjecture was unrelated to Fermat's conjecture, but surprisingly, Kennith Ribet and Gerhard Frey reasoned that Shimura-Taniyama conjecture doesn't hold unless there exists no positive integer solutions to $a^n+b^n=c^n$ for infinitely large $n$. In other words, Shimura-Taniyama conjecture begs the question that Fermat's last theorem is true. Hence, the Fermat's conjecture returned to mainstream of math proof not just for intellectual interest, but also for paving way for new branches of math research.   

\

Andrew Wiles, who has been obsessed with Fermat's Last Theorem since childhood, specialized in elliptic curves both as a graduate student in Cambridge and as a professor in Princeton. On hearing Ribet and Frey's research progress, he decided to prove Fermat's conjecture by attacking Shimura-Taniyama conjecture. He spent seven years working alone, learning and extending some new methods at that time,\cite{3} and presented his proof in 1993. While Wiles enjoyed public accolade and fame at first, a team of referees found in his paper a big flaw in a bound of the order of a specific group. \cite{4} Wiles spent one more year trying to repair his proof. Fortunately, before he wanted to give up an intuition struck him and help him correct the previous approach. Wiles published his manuscripts in 1995 on \textit{Annals of Mathematics}, and he proved Fermat's Last Theorem by techniques seemingly unrelated to number theory. After over 300 years of effort, mathematicians finally were able to prove Fermat's Last Theorem, and the process of proof is roundabout and interdisciplinary. 

In case anyone gets interested...

     Here is an article about the early history of Economics Department in University of Rochester. The author is its founder, Professor Lionel McKenzie. I got this paper from Professor Roman Pancs, instructor of grad level micro theory in U of R. 

      I am going to devote my lecture to a description of the early years of the Rochester department of economics. I think this attention to the early years is justified since the tradition of rigorous scholarship that has characterized Rochester economics over the years was established in that early period, and both faculty and graduate students, including Japanese graduate students, played important roles in this development. Indeed, an important part of the contribution of these people to economics was the tradition of rigorous scholarship that they began in this department.

     The Rochester Department of Economics was established in 1957 when I was appointed as its first chairman. I was chosen on the recommendation of Paul Samuelson and Robert Solow who were professors at MIT. The search for a chairman to undertake the establishment of a doctoral program in economics at Rochester had been underway for at least a year without success. Finally the Dean of the college Albert Noyes, a noted chemist, went to MIT to consult with Samuelson and Solow who recommended me. I was approached by Bernard Schilling, a professor of English from Rochester, who was on leave at Yale where I was visiting the Cowles Foundation, This led to a visit to Rochester to meet the selection committee. The two principal economists on this cornmittee were

Donald Gilbert and Robert France. Gilbert was acting chairman since the chairman William Dunkman was on leave at International Christian University in Japan. Gilbert was also Provost of the University. France had been appointed the previous year from Princeton, I was at Duke where I had been for nine years on returning from Oxford. After my visit I accepted the position at Rochester. I was attracted by the prospect of building a department according to my own ideas, which, in the event, would mean that economic theory would be the major emphasis. I was much impressed by the personality of Dean Noyes and accepted his assurances of the serious intention of the university to support the project. They also intended to develop graduate studies in other fields of the social

sciences and humanities. On the suggestion of Joseph Spengler, a colleague of mine on the Duke faculty, I asked for twenty graduate fellowships when the department reached its equilibrium size. The commitment was made and was kept As I told a later president of the University, Rochester never failed to fulfill its commitments to me. Previously there had been a Department of Economics and Business Administration. My first appointment was made even before the new department was

established. This was Ronald Jones on the recommendation of his thesis supervisor Robert Solow. Solow predicted, with perfect foresight that Jones would publish many good papers. Ialso quickly discovered that Ron was a fine lecturer, the type who gives a clean logical presentation without notes. My students will recognize this is not really my style. In the spring of 1957 Ron Jones and I visited Don Gilbert in Rochester. Unfortunately this would be our last contact with him since he died before I came into residence in the fall of 1957. Ron had to do some months of military service and did not

arrive until December of 1957. There were trvo holdovers from the old department of economics and business administration, in addition to France and Dunkman. These were Jack Taylor in industrial organization, and Warren Hunsberger in international trade. Neither of these quite fitted the objectives of the nerv graduate program and therefore were not retained. However, Jack Taylor remained in Rochester and taught for rnany years at St. John Fisher College. In the first year of the new department there were two visitors, J. N. Wolfe and Ian Macdonald. N{acdonald was a graduate student with me at Duke. Wolfe had been a student of John Hicks at Oxford and had been visiting in the University of Toronto. He ran our departmental seminar that year, His signal achievement was to persuade John Hicks to give a lecture in the spring. Hicks had been my supervisor at Oxford. On this occasion our first graduate student Akira Takayama took a famous snapshot of Hicks kneeling before our portable blackboard and writing mathematical expressions to illustrate his lecture on a Value and Capital theory of growth. This lecture was also given by Hicks in Berkeley, California, and somewhere in Japan. lt was heard by Roy Radner in Berkeley and by Michio Morishima in Japan. They immediately set to work to write important papers on the stability theory of optimal growth, which is often referred to as turnpike theory. My own contributions to this subject were not inspired by Hicks but by the book of Dorfman, Samuelson, and Solow. Morishima later lectured in Rochester, but I believe, unless my memory fails me, we were never visited by Radner. In his lecture Morishima confided to us that his mathematics was GI mathematics. since he learned it while lying in his bunk aboard Japanese naval vessels.

      Recruiting the new department began in the first year with two assistant professors Richard Rosett and Edward Zabel. Rosett was an econometrician from Yale who studied with James Tobin. I met him while visiting the Cowles Foundation in 1956. For me he was the most interesting participant in the Cowles coffee hour. Zabel was a theorist interested in theory of the firm and mathematical programming, who was with the Rand Corporation. He was recommended by Oskar Morgenstern With me, Jones, and France this completed the first faculty for the Ph, D. program, which offered the fields general equilibrium theory, theory of the firm, econometrics, international trade, and labor economics. The program was approved by the college, and seven students were

admitted and awarded fellowships for the academic year 1958-1959 One of these was Akira Takayama who had arrived the previous year, sent over by Don Gilbert's son in law, who taught at International Christian University in Japan. Five of the entering students succeeded in completing their degrees. Very appropriately they were distributed evenly over the early appointees. Akira Takayama worked with Jones with some advice from rne, Emmanuel Drandakis with me, William Bennett with Rosett, and Arnold SafIer with Zabel. Subsequently Takayama and Drandakis became successful teachers at the graduate level, Takayama in the United States and Japan and Drandakis in the United States and Greece. Takayama published a widely used textbook in mathematical

economics and Drandakis came to head the Athens School of Economics. Bennett became chairman of the economics department of the New York State College in Buffalo, and Saffer had a successful career in consulting for business firms. In the first years of the graduate program we were fortunate to have the example of intense application to study set by Takayama who would spend most of the night at his desk in the library. At the beginning of our program we received a small grant from the Ford

Foundation to provide support for summer research. However in a short while our young

assistant professors were able to get their own research funds from the NSF. This helped

to increase the interest in our department, at least in the States.

      

      When I refer to the early department Imean primarily the department during the first five years of the graduate program from 1958-59 to 1962-63. I will show you a departmental picture taken in the spring of 1962. This picture hangs in our common room in Rochester. There are two later pictures of the department hanging in the common room, but only in this one do the members of the department wear jackets and ties. So has academic dress declined, at least in Rochester. However let me add that for the other half of the original Department of Econornics and Business Administration, who developed into the Simon School of Business, jackets and ties are expected In the early days I

functioned as a benevolent dictator since the new appointments who were at the core of the graduate program did not have tenure. The department eventually choose Dick Rosett to suggest to me that it might be a good idea to hold departmental meetings. Apparently only Rosett had the nerve to approach the dictator. Iclaim that I readily agreed to the meetings. I may add that in those days rve kept minutes of our meetings a practice that later lapsed well after the end of my chairmanship. Alater appointment, Sherwin Rosen who became chairman of the Chicago department of economics, told me that he admired the way I would introduce the agenda so that I always got my preferred decision adopted. He said that he later tried to follow my example at Chicago but it didn't work.

Perhaps you will forgive me for introducing some lighter subjects that the members of the early department like to recall. Bob Fogel has described a practice which was quite unknown to me. Apparently a few members of the junior staff sometimes met in one of their oflices for card games, either bridge or poker. Since I was a benevolent dictator in those days they kept a sharp lookout for me, apparently thinking I might take vengeance on them for such an unproductive use of their time. Although I doubt that their concern was justified they did succeed in keeping the card game secret from me. However, another fun activity included me. We had a bowling team in the University6

league that we called "The Variables", Ibelieve the name was due to Dick Rosett. The team included me, Jones, Rosett, and sometimes Penner. I'm afraid we were not very good, partly because the best bowlers in the department,EdZabel and Bill Dunkman were already on other teams. There was also a group of golfers who took advantage of a public golf course across the road from the University in Genesee Valley Park. it included Ron Jones, Ed Zabel, Bill Dunkman, and later Sherwin Rosen, comprising a foursome. Sometimes graduate students would join them. In addition to these activities

being a young department with attractive wives we had many parties. Also a nice occasion was the tea in the faculty club that the department gave the graduate students at the beginning of each academic year. The wives served the tea from silver teapots provided by the University Women's Club, and alcohol was not served. The conversations between faculty and students were quite lively. Alcohol intruded later and in my opinion the event was never as successful after that happened. We also started the custom of a departmental picnic which has continued to the present day, though in our

day athletic contests were more popular among those at the picnic. With regard to the social environment of the early department a special mention is merited of my wife Blanche. She played a large role in making both new faculty and students comfortable in their new environment. She continued to do this throughout her life but it was especially important for the early department. In 1995 at a reception after I had received the medal of the Rising Sun Akihiro Amano in a tribute to her referred to her as their academic mother.

      The association of Rochester economics with Japan was already well established

in these early years by a series of outstanding students who completed the Ph. D."l

program.. Iwill say a few words about them. As I mentioned Akira Takayama was in

residence when the doctoral program began. Both Akira and Ron Jones attended my first

value theory course in the spring of 1958. Akihiro Amano entered in 1960. He was

planning to stay for only one year but he found our program attractive enough that he

decided to remain for the full Ph D. program. Working with Ron Jones he produced an

excellenthesis in the field of international trade. In particular, Amano introduced a new

notation that facilitated the proof of trade theorems. In 1961 another Japanese student

Hiroshi Atsumi entered. He was recommended to us by his professor in Japan, Hukukane

Nikaido. Professor Nikaido, rvho rvas at Osaka University at that time, was one of the

economists whom we had invited to send students to Rochester. Atsumi wrote his thesis

with me. It was very original and made important contributions to the theory of optimal

growth. Together with the German economist Christian Weizsacker he introduced the

overtaking criterion in the theory of optimal growth. This allowed him to deal with

problems in which the horizon is taken at infinity. He was also the first theoristo

produce an optimal growth model with more than one sector. According to Jones and

Rosett, when their classes became boring Hiroshi would turn offhis hearing aid.

However I never observed this behavior mvself. Nobuo Minabe entered in 1961 anrl

worked with Ron Jones. He made extensions of international trade theory by use of

geometrical methods. Finally Yasuo Uekawa entered in 1963. We believe the students

from Kansai probably learned of Rochester from Professor Mizutani at Kobe Commercial

University. Mizutaniwas a friend of Ed Zabel. Following a suggestion from the visiting

professor John Chipman, Uekawa generalized in significant ways Stolper-Samuelson

theory in international trade and the theory of factor price equalization between countries8

that enjoy free trade. It is sad to record that Takayama and Uekawa are deceased. The

economics department has established fellowships in memory of them for which

Japanese students are given preference, Of the four Japanese graduates who were

admitted by 1963 three subsequently taught at Rochester as visitors. These are Takayama,

Amano, and Atsumi. Altogether 23 students who completed their doctorates were

admitted in these years. In addition to students from Japan there were students from the

United States, Greece, India, and Taiwan. It was our policy to admit and support students

purely on the basis of their ability with no attention to their country of origin. On one

occasion the State Department refused to admit a Taiwanese student because he had

family in communist China and therefore might be subjecto intimidation. Ithreatened to

resign over the issue and got the support of our congressman. Happily they relented.

Initially our graduates had some difficulty obtaining jobs in some countries from which

they had come. This was never true of Japan but discrimination seemed to be practiced in

India. However, today India no longer discriminates. Indeed I have sometimes asked

people what university has had the largest number of Rochester graduates teaching at one

time. They never guess that it is Calcutta with nine.

       In the Fall of 1962 two appointments to the University were made that had

considerable implications for our department in future years, William Riker was

appointed with the charge of developing a Ph. D. program in political science, and Allen

Wallis was chosen to be president of the University. Wallis was recommended to us and

to the selection committee by Milton Friedman. Friedman was brought to Rochester to

appear in a lecture series that had been established in honor of Donald Gilbert. I recall my

discussion with Bill Riker when he came to the University to consider whether to accept9

a position there. He had spent some time at the Center for Advanced Study in the

Behavioral Sciences in Stanford in the company of Kenneth furow. Thus he was

interested in applying analytical methods to political science. Of course I welcomed him

heartily to Rochester and described my department's goals to him. The presence of an

economics departrnent whose success was already well known played an important role

in making these appointments possible.

      We did something to cement relations with political science by appointing

Duncan Black as a visitor for 1962-3 directing him to the political science department.

Duncan Black was an early contributor to the analysis of decision making by committees.

This was close to the interests of Riker who was applying game theory to competition

between political parties. He was also an expert on the mathematician, Lewis Carroll, the

author of Alice in Wonderland and an early theorist of political choice in committees and

elections. Wallis developed a new businesschool with a major emphasis on economics.

Although we remained independent of the business school, over the years there has been

considerable interaction between their economists and ours, especially in the area of

maaroeconomics. There were many appointments that were joint both with political

science and the business school. Indeed, at one time I believe a large lraction of the

members of the political science department held Ph. D.'s in economics.

      After the first year of the graduate program several additional appointments were

made to the early department. Nanda Choudry was appointed in 1959 as an

econometrician. He later went to Toronto. Richard Weckstein came in 1959 as a visitor to

teach development economics. He later went to Williams College. In 1960 S, C. Tsiang

was appointed. This allowed the macroeconomic field to be established. S. C. Tsiang hadl0

been teaching at Johns Hopkins University and serving as an adviser to the government

of Taiwan. He also bears some responsibility for the monetary stability of the economy of

Taiwan. We were warned that Tsiang was not very effective as a teacher of large

undergraduate classes. However, this did not deter us. He taught graduate and

undergraduate seminars very well for us. Harry Grubert from MIT was also with us for a

few years in the macro field. He is now in the United States Treasury with responsibilities

for foreign transactions. Alexander Eckstein, a specialist in the economy of communist

China, was appointed in 1960. He brought the distinguished Japanese economist Shigeto

Tsuru to Rochester as a visitor. My wife and I enjoyed our first Japanese dinner in his

apartment. In 1961 Rudolph Penner was appointed for the field of public finance, which

had been a special interest of the former chairman, Donald Gilbert. Penner later became

head of the Congressional Budget Oflice in Washington, D. C. In 1963 the field of

econometrics received a major boost from the appointment of Michio Hatanaka who had

been working with Oskar Morgenstern in Princeton and played a role in introducing

spectral analysis to econometrics. Morgenstern had been the supervisor of my graduate

study at Princeton.. Norman Kaplan replaced Eckstein in 1963. Kaplan was an expert on

the Soviet economy Both Eckstein and Kaplan held the Xerox professorship supported by

the Xerox Corporation, a professorship which is now held by Ronald Jones.

      After the first year of the graduate program my proudest recruiting was done to provide strength in economic history and labor economics. Our department had taken as its mission the promotion of theory and its utilization in applied fields. We wanted to find people who could use theory, mathematics, and statistics in labor economics and economic history. To say the least we were either extremely clever or extremely lucky.tl tl Alexander Eckstein soughthe advice of the economic historian Simon Kuznets at Johns Hopkins University. Kuznets recommended Robert Fogel. We heard Fogel in seminar one day and appointed him the next day You could do that in those days This was in face of the fact that he had been a member of the Communist Party and had a rnixed marriage. These things mattered in those days. But we regarded the prejudices of other people as our opportunities. He gave us his paper on the prospects of the Union Pacific Railroad at its inception, and we had no doubt that he met our requirements. At Rochester he did his research on the canal system in the United States in the period of railroad building for which he received the Nobel Prize.In making this appointment I had the strong support of Dick Rosett. Fogel reported his research to a meeting of the American Economic Association. Akihiro Amano told me recently that he and another student did the final preparation of Fogel's statistical materials in our primitive econometrics

laboratory, leaturing only Marchant calculators, while Fogel waited impatiently outside

in his car to leave for the airport. Amano remarked that thus was cliometrics born at Rochester. Bob Fogel has been fond of telling a story to illustrate the spirit of the early department. He asked Ron Jones for an introduction to general equilibrium theory which led Ron to give him a series of private lectures on general equilibrium This in turn led to an important paper by Jones on a simple model of general equilibrium which was published in the Journal of Political Economy. Fogel also received instruction in econometrics from Dick Rosett. When the Nobel Prize was awarded Fogel had moved to Chicago, so the fact that he had done the work at Rochester was not always made clear.

The recruitment of our labor economist was just beyond the early department by my definition. Bob France approached the Chicago department \n 1964 with ourt2 specifications for this appointment and received a recommendation for Sherwin Rosen.

      Sherwin had the mathematical background of an electrical engineer. There were some

reservations in the recommendations from Chicago but we felt we knew better. As it

turned out we were right. Rosen played a major role in bringing economic theory to bear

on labor economics and would with high probability have received the Nobel prize

except for his untimely death from lung cancer. The program in economic history was

further strengthened by the appointment in 1963 of Stanley Engerman, also a student of

Simon Kuznets. Engerman came to Rochester first as a replacement for Bob Fogel who

went to visit Chicago, but the arrangement turned out to be permanent, although Fogel

later returned on halftime basis.A famous research program was undertaken by Fogel and

Engerman comparing the productivity of slave laborers in the southern states of the

United States with the free laborers of the northern states. They showed that the slave

workers compared favorably with the free workers in productivity though, of course, they

did not condone the institution of slavery.

      Thus the early department had distinction in economic theory, international

economics, and comparative economic systems. Also the basis was laid for achieving

distinction in the fields of econometrics, labor economics, and economic history.

Macroeconomics remained rather weak since the expertise of S. C. Tsiang was

principally in the theory of money, although the appointment of Hugh Rose in 1961, on

the recommendation of Harry Johnson, added strength to this field. It is interesting that in

the 1970's with the appointment of Karl Brunner, Robert Barro, and Robert King,

macroeconomics became a very strong field, perhaps the field for which the deparlment

was best known in the United States in that period. This strength has continued to thel3

present day. In later years a notable change that occurred in theory was the rise of game

theoretical rnethods and, beginning with James Friedman, Rochester has reflected the

change. It should also be mentioned that we have had a series of very distinguished

econometricians succeeding Hatanaka, beginning with G. S. Madala.

       Of the five professors who initiated the graduate program in 1958 Jones and I

have remained in the Rochester economics department to this time, although Ihave not

taught classesince 1995. Zabel left to go to the University of Florida in Gainsville in

1980 and is now deceased. Rosett left Rochester in 1974 to become Dean of the Graduate

School of Business in the University of Chicago and later returned to Rochester to be

Dean of the College of Business in the Rochester Institute of Technology. He was also at

one time Chairman of the board of directors of tlie National Bureau of Econornic

Research France became Director of Budgets for the University and Vice-President for

Planning. He is now decease

Discover Quality Change of U.S Postsecondary Institutions from 1990 to 2010 (conclusion and further consideration)

      In this paper I mainly used expenditure records of four types of postsecondary institutions to discover where schools have been investing in. I used fixed effect (FE) model regression to find out contributing effects of investment on instruction and research and inferred quality change of higher education. The results are that private institutions and public research institutions all experienced some levels of quality improvement while community colleges are left behind.

      There are several advantages of this approach. Firstly, expenditure data are relatively complete and accessible to general public[1]. In addition, using expenditure per FTE student as a measure of quality avoids particular assumptions about the best, or preferred, way for schools to allocate their total resources among specific inputs. (Ladd and Loeb, 2012) For example, an institution can use a certain amount of instruction per FTE student either for smaller classes with less experienced instructors or larger classes with more experienced instructors. With no evidence showing that certain categories of resources are preferred to others in all postsecondary institutions, it would be inappropriate to conclude that one way of spending is absolutely better than the other. Finally, this measure allows for straightforward comparisons across schools, and by using regression tests, researchers can also study how one category of spending affects total expenditure.

      However, there are three main limitations with this method. Firstly, different schools report data under various accounting standards, which creates debate upon what data reports to use. Besides, we can only draw an inference of how quality changes by discovering where money has gone and whether institutions are putting more money in core missions like instruction. Lastly, this method takes no account of differences in effectiveness or efficiency with which dollars are spent. The key assumption of input measure is that high quality inputs tend to produce better education provided that market is competitive. We don’t know if students will fully utilize inputs deemed to be of high quality, such as whether students will have more contact with professors or do more critical thinking. Even though higher education market is generally competitive (Hoxby 1997), there would still be concern about inefficiency given the unique governance of institutions.[2]

      Since data on mean SAT scores are incomplete, I couldn't create a quality index to give numerical assessment of quality, which is another popular input measure methodology. Two papers can exemplify and reveal questions about this approach. Black and Smith (2006) used index as proxy of college quality to discover its impact on graduates’ income. They first summarized previous related researches and then compared four measures with multiple proxies. They found that Generalized Method of Moments (GMM) estimation is the most efficient econometrics model and proxies that are powerful are mean SAT scores, mean faculty salaries, student-faculty ratio, application rejection rate and first year retention rate. Cohodes and Goodman (2013) criticized that Black and Smith’s choice of proxies is partly flawed because of certain level of perfect correlation. Instead, they introduced another proxy, namely college completion rate, into their index. One obvious problem with this approach is that quality index is created rather arbitrarily: there is no consensus about what proxies should be chosen and what weights should be assigned. What’s more, most researchers use index to function as a regressor for graduates’ income. In other words, quality of education is not their major concern.

      Recently there are some new approaches trying to do a better job. For example, Porter (2012) argued that detailed student self-report can help discover what they have obtained in college. This attempt is a value-added measure and is theoretically desirable.  However, the validity of this approach depends on whether reports are well-designed. 

      Another attempt is more promising. With the advent of online course platforms like Massive Open Online Course (MOOC) system, people are able to learn college materials without actually sitting in a classroom. I think it possible to design a value-added like measure. We can do comparison tests between two groups of students with similar abilities and motivations and have them take several courses. One group learns materials in class, while the other takes online versions. The main control is that the former group is accessible to all those services related to instruction while the group that takes online versions is not. At the end of the semester both groups take the same tests and we compare their test results. If students who learn materials in class do perform better, then we might conclude that sitting in class and enjoying services like academic support can help improve students’ performance; then we can step further and test to what extent does these services help and find out possible inefficiency in terms of expenditure. This method is not impeccable, however, since we might also count peer effect as positive effect of instruction related spending.

      As Robert Pirsig (1974) wisely wrote, “Some things are better than others; that is, they have more quality. But when you try to say what the quality is, apart from the things that have it, it all goes poof. ” Quality measure in terms of higher education is hard because many aspects of college life cannot be quantified, and there are a lot of factors like student motivation that affects student performance. We might never know how college undergraduate education really bestows on us, and all we can do is come up with some ways to get a better approximation. 

references:

1.      Ladd, Helen and Susanna Loeb. “The Challenge of Measuring School Quality: Implications for Educational Equity,” Education, Justice and Democracy, The University of Chicago Press, pp. 22-55, forthcoming

2.      Black, Dan and Jeffrey Smith. “Estimating the Returns to College Quality with Multiple Proxies for Quality.” Journal of Labor Economics, Vol. 24, No. 3, 2006

3.      Cohodes, Sarah and Joshua Goodman. “Merit Aid, College Quality and College Completion: Massachusetts’ Adams Scholarship as an In-kind Subsidy.” American Economic Journal: Applied Economics, forthcoming, March 2013

4.      Pirsig, Robert. Zen and the Art of Motorcycle Maintenance: An Inquiry into Values. Harper Torch; Reprint edition (April 25, 2006)  ISBN-10: 0060589469

 

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[1] Still, some specific information like investment in each department over years is unavailable.

[2] Ronald Ehrenberg elaborated possible reasons why American postsecondary institutions might suffer from inefficiency in Rising Tuition. (ISBN-13: 978-0674009882)