Hedonism

Instrumental goods: things that are valuable because of the good things they bring about.
wa
Intrinsically valuable goods: things whose goodness is self contained and valuable in its own right, even if they bring nothing else in its wake.

Happiness is intrinsically valuable. In fact, for hedonists, this is the only thing that is intrinsically valuable, and unhappiness is the only thing that directly reduces our quality of life.

According to hedonists, a life is good to the extent that it is filled with pleasure and is free of pain.

Two kinds of pleasure:
(1) physical pleasure
(2) attitudinal pleasure (enjoyment)

Hedonists believe that happiness is attitudinal pleasure. Hedonism should be understood as the view that enjoyment, rather than physical pleasure, is the key to the good life.

Epicurus argued that the most pleasant condition is one of inner peace. The ideal state of tranquility comes largely from two sources: moderation in all physical matters, and intellectual clarity about what is truly important.

John Stuart Mill argued that pleasures come in different levels of quality, and that the best pleasures for human beings were those that come only through hard work. Hedonism explains why there are many ways to live a good life: there are many paths to happiness. What makes one person happy is largely a matter of his personal choice. However, hedonism does not allow us to have the final say about what is good for us.

Critiques of hedonism:
(1) The paradox of hedonism
If something always makes us better off, then it seems reasonable to try very hard to acquire it. With happiness, however, this might backfire--those who try real hard to make themselves happier almost never succeed. This is the paradox of hedonism. Let's see the following argument:
(a) If happiness is the only thing that directly makes us better off, then it is rational to single-mindedly pursue it.
(b) It isn't rational to do it.
(c) Therefore, happiness is not the only thing that directly makes us better off.

The argument is valid, but probably not sound. The bottom line is even if happiness is all that matters, it may not be rational to aim for it directly. (which means premise (a) is false)
So this objection to hedonism is not sound.

(2) Evil pleasures
Some people are happy to do awful things. Argument:
(a) If hedonism is true, then happiness that comes from evil deeds is as good as happiness that comes from kind and decent actions.
(b) Happiness that comes from evil deeds is not as good as happiness that comes from kind and decent actions.
(c) Hedonism is false.

However, there is confusion in this statement, which undermines its validity.
Why? According to hedonists: happiness gained from evil deeds can improve our lives just as much as happiness that comes from virtue. Thus, hedonists reject premise (b).

(3) The two worlds (W.D. Ross)
Consider two worlds that contain identical amounts of happiness and misery. In one of these, the people are all virtuous; in the other, they are all vicious. According to hedonism, the two worlds are equally good. Argument:
(a) if hedonism is true, then any two situations containing identical amounts of happiness and unhappiness are equally good.
(b) some such situations are not equally good; some are better than others.
(c) hedonism is false.

Hedonists reject premise (a). Their view is not about what makes a situation or a world good, but about what makes a life good for the person who lives in it. Hedonism, thus, doesn't tell us how to determine the value of the world. So it is not committed to the view that two worlds containing equal amounts of happiness must be equally good.

(4) False happiness
(a) If hedonism is true, then our lives go well to the extent that we are happy.
(b) It is not the case that our lives go well to the extent that we are happy; those whose happiness is based on false beliefs have worse lives than those whose happiness is based on true beliefs, even if both lives are equally happy.
(c) hedonism is false.

This claim assumes that the source of happiness determines how beneficial that happiness is. But hedonists deny this. For them, happiness is happiness. However, a pleasant life of illusion is less good for you than an equally pleasant life based on real achievement and true beliefs.

(5) autonomy
We want autonomy--the power to guide our life through our own free choices---even if it sometimes costs us our happiness. Preserving our autonomy is vitally important, even if it doesn't always make us happier.

Paternalism: someone's limiting your liberty against your will, but for your own good.

Argument:
(a) if hedonism is true, then autonomy contributes to a good life only insofar as it makes us happy
(b) autonomy sometimes directly contributes to a good life, even when it fails to make us happy
(c) hedonism is false

The argument is legit.

(6) Life trajectory
(a) if hedonism is true, then the overall quality of life depends entirely on the amount of happiness and unhappiness it contains
(b) the overall quality of life depends on at least one other factor: whether one's life reflects an "upward" or "downward" trajectory
(c) hedonism is thus false

seems good

(7) Unhappiness as a symptom of harm
Argument from multiple harms
(a) if hedonism is true, then you can be harmed by something only because it saddens you
(b) you can be harmed in other ways
(c) hedonism is false
Hedonism fails to appreiciate that unhappiness is often a symptom that something intrinsically valuable--something other than happiness--has been lost.

Notes of Introductory

The essay is about the nature and limits of the power which can be legitimately exercised by society over the individual.

Tyranny of the majority. Protection against the tyranny of the magistrate is not enough; there needs protection also against the tyranny of the prevailing opinion and feeling. There should be a limit to the legit interference of collective opinion with individual independence.

The question is how to find the limit.

The principle of the essay: the sole end for which mankind are warranted, individually or collectively, in interfering with the liberty of action of any of their number, is self-protection. That the only purpose for which power can be rightfully exercised over any member of a civilized community, against his will, is to prevent harm to others. Over himself, over his own body and mind, the individual is sovereign.

Notice that the above principle is meant to apply only to human beings in the maturity of their abilities. As for children, they must be protected against their own actions as well as against external injury. Liberty, as a principle, has no application to any state of things anterior to the time when mankind have become capable of being improved by free and equal discussion.

Three basic liberties:
(1) liberty of conscience, thought and feeling;expressing and publishing opinions.
(2) liberty of tastes and pursuits
(3) liberty of combination among individuals (persons combined are full age, and not forced or deceived)

The only freedom which deserves the name, is that of pursuing our own good in our own way, so long as we do not attempt to deprive others of theirs, or impede their efforts to obtain it.

ticking

In this paper, the author Lafollette argued that parenting should be licensed.

Lafollette first discussed two criteria for regulation:
(1) the behavior can be potentially harmful to others
Example: driving recklessly leads to injury and fatality, so driving is licensed
(2) competence is required
Example: lawyers, pshychiatrists, drivers need skills

In terms of parenting, bad parents can do great harm to children, so the first criterion is satisfied. As for competence, parenting does need skills and practice.

Lafollette believed that the only ways to deny the necessity of parenting regulation are to
(1) deny licensing any potentially harmful activity;
(2) deny the two criteria of regulation
(3) deny that parenting satisfies the two criteria, assuming the validity of the criteria
(4) show that even though parenting satisfies the two criteria, there exist special reasons that licensing parents isn't good
(5) show that there is no reliable and just procedure for implementing regulation

(1)(2)(3) are trivial, the author focused on (4)(5)

Theoretical objections to licensing:
(1) people have free right to have children, which shouldn't be licensed.

Lafollette's opinion: even if people have right to have children, the right might be limited to protect innocent people, i.e. children. Parents' way of rearing children might be unacceptable even if they consider it fit. He believes that a person has a right to rear children if he meets certain minimal standards of child rearing. Denying a parenting license to someone who is not competent does not violate that person's rights. Rights are conditional.

The purpose of licesing is to prevent serious harm to children, and prior restraint required by licensing would not be very onerous for many people.

(2) a parental licensing program would deny licenses to applicants judged to be incompetent even though they had never maltreated any children (prior restraint)

The author argues that individuals denied license are given the opportunity to reapply easily and repeatedly for a license. [take counseling or therapy to improve their chances of passing the next test]

The author reasoned that: even though one needs to be worry about prior restraint, if the potential for harm is great and the restraint is minor relative to the harm we are trying to prevent, then such restraint is justified. 

My thoughts:

Practical objections to licensing:
(1) what are adequate criteria of "a good parent"? Knowledge problem

Author: the proposal is only to exclude the very bad parents.

My thoughts: in terms of excluding bad parents, probably only the extreme cases are to be distinguished. What about other cases?

(2) there is no reliable way to predict who will maltreat their children

Author: Other licensing programs are not predict 100% accurately, so we cannot demand more in terms of parental regulation. We can use existing tests that claim to isolate relevant predictive characteristics that signifies bad parents to find potentially bad parents. He also believes that tests will be improved.

My thoughts: just because the other similar regulation is not perfect doesn't mean you can adopt this system into parenting.

(3) administrators might unintentionally misuse that test, which would harm innocent individuals.

Author: the fact that mistakes are made shouldn't lead us to abandon attempts to determine competence.

My thoughts: the same as before. In addition, to get pass the test, individuals might use other ways to cheat.

(4) any testing procedure will be intentionally abused.(administrators' self interest)

Author: there is no reason to believe that the licensing of parents is more likely to be abused than drivers' license tests or other regulatory procedures. In addition, individuals can appeal.

My thoughts: red-tape? bureaucrats' self-interest?

(5) It is hard to implement this program

Author: if it is important enough to protect children from being maltreated by parents, then surely a reasonable enforcement procedure can be secured.

My thoughts: costs?

1/14/2015

Value theory: What is the good life? What's worth pursuing for its own sake?

Normative ethics: What're our fundamental moral duties? What counts as virtues, vices and why?

Metaethics: What's the status of moral claims and advice?

Certainly, many laws require what morality requires, and forbid what morality forbids. But the fit is hardly perfect, and that shows that morality is something different from the law. Good manners, similarly, are not the same thing as morally good conduct. The same is true when it comes to self-interest or tradition. In terms of tradition, people sometimes speak of conventional morality, which is the set of traditional principles that are widely shared within a culture or society. However, some social standards and mutual endorsement can be morally mistaken. As a result, the moral we refer to in this context are standards that are independent of conventional morality and ca be used to critically evaluate its merits. This implies the existence of some independent, critical morality that
(1) doesn't have its origin in social agreements
(2) is untainted by mistaken beliefs, irrationality, or popular prejudices
(3) can serve as the true standard for determining when conventional morality has got it right and when it has fallen into error.

How to distinguish between valid and invalid arguments.
(1) identify all of an argument's premises
(2) imagine that all of them are true
(3) ask yourself: suppose all premises were true, could the conclusion be false? If yes, the argument is invalid. If no, the statement is valid.

But a valid argument is not enough, because in testing the logical structure we assumed the truth of premises, which might not be the case. What we need is a sound argument, and that means the logic is valid and premises are true.

Example:
premise: it is morally acceptable for nonhuman animals to kill and eat other animals
conclusion: it is morally acceptable for human beings to kill and eat other nonhuman animals.

the premise doesn't logically support the conclusion. (What is morally acceptable for animals is not necessarily morally acceptable for us)

This shows that this particular way of defending is no good.

A valid argument:
premises:
(1)if it is morally acceptable for nonhuman animals to kill and eat another, then it is morally acceptable for humans to kill and eat other nonhuman animals.
(2)it is morally acceptable for nonhuman animals to kill and eat other animals
conclusion:
it is morally acceptable for human beings to kill and eat other nonhuman animals

The argument is valid but the first premise is wrong. Animals are not moral agents--they cannot control their behavior through moral reasoning, and thus it is not convincing to look to animals for moral guidance.

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