Consistency and Convergence

A consistent sequence of estimators is a sequence of estimators that converge in probability to the quantity being estimated as the index (usually the sample size) grows without bound. In other words, increasing the sample size increases the probability of the estimator being close to the population parameter. Mathematically, a sequence of estimators  is a consistent estimator for parameter θ if and only if, for all ε > 0, no matter how small, we have

 

The consistency defined above may be called Weak Consistency. The sequence is Strongly Consistent, if it Converges Almost Surely to the true value. To say that the sequence X_n converges almost surely or almost everywhere or with probability 1 or strongly towards X means that

This means that the values of X_n approach the value of X, in the sense (see almost surely) that events for which X_n does not converge to X have probability 0. Using the probability space and the concept of the random variable as a function from Ω to R, this is equivalent to the statement

Indirect Least Squares Method (ILS)

Suppose we wish to estimate a structural equation containing say, three endogenous variables. The first step of the ILS technique is to estimate the reduced-form equations for these three endogenous variables. If the structural equations for these three endogenous variables. If the structural equation in question is just identified, there will be only one way of calculating the desired estimates of the structural equation parameters from the reduced-form parameter estimates. The structural parameters are expressed in terms of the reduced-form parameters, and the OLS estimates of the reduced-form parameters are plugged in these expressions to produce estimates of the structural parameters. Because these expressions are nonlinear, however, unbiased estimates of the reduced-form parameters produce Only Consistent estimates of the structural parameters, not unbiased estimates.

If an equation is over-identified, the extra identifying restrictions provide additional ways of calculating the structural parameters from the reduced-form parameters, all of which are supposed to lead to the same values of the structural parameters. But because the estimates of the reduced-form parameters do not embody these extra restrictions, these different ways of calculating the structural parameters creates different estimates of these parameters. (This is because unrestricted estimates rather than actual values of the parameters are being used for these calculations.) Because there is no way of determining which of these different estimates is the most appropriate, ILS is not used for over-identified equations. The other simultaneous equation estimating techniques have been designed to estimate structural parameters in the over-identified case; many of these can be shown to be equivalent in the over-identified case; many of these can be shown to be equivalent to ILS in the context of a just-identified equation, and to be weighted averages of the different estimates produced by ILS in the context of over-identified equations.

Here is a basic procedure to implement ILS:
1. Rearrange the structural form equations into reduced form, Estimate the reduced form equations;
2. Estimate the reduced form parameters;
3. Solve for the structural form parameters in terms of the reduced form parameters, and substitute in the estimates of the reduced form parameters to get estimates for the structural ones.
Note: If structural equation is exactly identified, there will be  a unique way to calculate the parameters. Estimates of reduced form parameters are unbiased, but estimates of the structural parameters will not be.  Both are consistent.

Order & Rank Conditions of Identification

The identification problem is a mathematical (as opposed to statistical) problem associated with simultaneous equation systems. It is concerned with the question of the possibility or impossibility of obtaining meaningful estimates of the structural parameters. The identification problem can be solved if economic theory and extraneous information can be used to place restrictions on the set of simultaneous equations. These restrictions can take a variety forms (such as use of extraneous estimates of parameters, knowledge  of exact relationship among parameters, knowledge of the relative variances of disturbances, knowledge of zero correlation between disturbances in different equations, etc.), but the restrictions usually employed, called Zero Restrictions, take the form of specifying that certain structural parameters are zero, i.e., that certain endogenous variables and certain exogenous variables do not appear in certain equations. Mathematical investigation has shown that in the case of Zero Restrictions on structural parameters each equation can be checked for identification by using a rule called the Rank Condition. It turns out, however, that this rule is quite awkward to employ, and as a result a simpler rule, called the Order Condition, is used in its stead. This rule only requires counting included and excluded variables in each equation.

Here is a brief illustration of order and rank conditions of identification in simultaneous equation system:

M = number of endogenous variables in the model
K = number of exogenous variables in the model
m = number of endogenous variable in an equation
k = number of exogenous variables in a given equation
Rank condition is defined by the rank of the matrix, which should have a dimension (M-1), where m is the number of endogenous variables. This matrix is formed from the coefficients of the variables (both endogenous and exogenous) excluded from that particular equation but included in the other equations in the model.
The rank condition tells us whether the equation under consideration is identified or not, whereas the order condition tells us if it is exactly identified or overidentified.
1. If K-k>m-1 and the rank of the ρ(A) is M-1 then the equation is overidentified.
2. If K-k=m-1 and the rank of the ρ(A) is M-1 then the equation is exactly identified.
3. If K-k>=m-1 and the rank of the ρ(A) is less than M-1 then the equation is underidentified.
4. If K-k<=m-1 the structural equation is unidentified. The rank of the ρ(A) is less M-1 in this case.

From these rules, we can tell that, the order condition is only a necessary condition, not a sufficient one. So that, technically speaking, the rank condition must also be checked. Many econometricians do not bother doing this, however, gambling that the rank condition will be satisfied (as it usually is) if the order condition is satisfied. This procedure is hence not recommended.

Direct PC SAS Output to a File

When running SAS programs interactively through the display manager, the output from any procedure is written to the Output window and notes, warnings and errors are written to the Log Window. Contents of these windows are temporary. They can be saved to a file using the File Save pulldown menus from the Output Window and from the Log Window. But if you want to make sure that the output of these windows is saved to a file every time, you can use Proc Printto to automatically route output to a file.

For example, the following program routes the output from Proc Printto directly to a file named auto.lst. What would have gone to the Output Window is redirected to the file c:\auto.lst . The statements below tell SAS to send output that would go to the Output Window to the file c:\auto.lst and to create a new file if the file currently exists.  If the NEW option was omitted, SAS would append to the file if it existed.

    PROC PRINTTO PRINT='c:\auto.lst' NEW;
    RUN;

Note: (1) sometime SAS program can collapse (unexpected terminated) before it executes all of the statements properly, then you will lose all of the results you already got.(In this kind of situation you have to end the SAS through Windows Task Manager, because generally the SAS program will stop respond). By using Proc Printto, you can save all of the temporary results you have already got before the program is unexpected terminated.

(2) Generally you need put Proc Printto statement at the very beginning of the SAS code. Of course, you can also release the print output file by using another simple statement at the very end of the SAS code:
    PROC PRINTTO;
    RUN;

(3) For log print, you can use a similar SAS Code:

    PROC PRINTTO LOG='c:\auto.log' NEW;
    RUN;

Observations and Thoughts on Haiti and Chile

Here are some observations from a blogger:
"The recent earthquakes in Haiti and Chile present an interesting contrast between the deleterious effects of a major earthquake in one of the richest countries in the western hemisphere and in the poorest.  It may surprise you that Chile is (by relative standards) quite an advanced and relatively wealthy country as many Americans, I think, have a tendency to view all of Latin America as a poor region.  According to the CIA, the per-capita GDP in Chile in 2009 was $14,700 while Haiti was $1,300 - so while Chile is far from US or Western European standards of living, it is a much wealthier country than Haiti.  In both cases the earthquake (and subsequent tsunami in Chile) were devastating disasters, but the scope of the tragedy in Haiti was, it appears, much, much worse."

These observations pass two serious thinking to me:
1. Other than physical demand, people's level of immaterial demand can also be determined by income or wealth; and most of time, safety is not among the basic levels of human needs.
2. Opportunity cost for poor is less than rich people when they are facing the same danger and potential of losing. Who can stand more risk and unsafety, poor or rich? This is a two-way argument.

So the practical question is that, can we validate these observation via some statistical or econometrics methods?

A good Illustration of Weighted Regression by Peter Kennedy

Measurement Error

In parametrics, the assumption of fixed regressors is made mainly for mathematical convenience, if the regressors can be considered to be fixed in repeated samples, the desirable properties of the OLS estimator can be derived quite straightforwardly. The essence of this assumption is that, if the regressors are nonstochastic, they are distributed independently of the disturbances. If this assumption is weakened to allow the explanatory variables to be stochastic but to be distributed independently of error term, all the desirable properties of the OLS estimator are maintained; their algebraic derivation is more complicated, however, and their interpretation in some instances must be changed (for example, in this circumstance, βOLS is not, strictly speaking, a linear estimator).

If the regressors are only contemporaneously uncorrelated with with the disturbance vector, the OLS estimator is biased but retains its desirable asymptotic properties at the expense of the small-sample properties of βOLS. If the regressors are contemporaneously correlated with the error term, the OLS estimator is even asymptotically biased.

When there exists contemporaneous correlation between the disturbance and a regressor, alternative estimators with desirable small-sample properties can not in general be found; as a consequence, the search for alternative estimators is conducted on the basis of their asymptotic properties. The most common estimator used in this context is the instrumental variable (IV) estimator.

The Nature of Agricultural Economics - My Thought

The strength of agricultural economics is not that it can compete with general economics research. People may feel that general economics research is more decent, this is true in the sense that it produces pretty neat and nice work with help of mathematic notation. Mathematics is important, this should be admitted, it is the logic language of this world. So what general economics research does is that it has been employed in the effort of expressing the world, while the agricultural economics research should be dedicated to be more close to the real world, to pass more care to people and entire world's basic needs.

There is a movie which has been putting on the screen for a while, Food Inc. (2008). An American documentary film directed by Emmy Award-winning filmmaker Robert Kenner. The film examines large-scale agricultural food production in the United States, concluding that the meat and vegetables produced by this type of economic enterprise have many hidden costs and are unhealthy and environmentally-harmful. The documentary generated extensive controversy in that it was heavily criticized by large American corporations engaged in industrial food production. This is just an example, so the question is that after popular cost-benefit analysis and its derivative forms and combinations, who really cares about problem like above? Most of time, optimization, maximization, equilibrium and so forth are too perfect to be practical in applications; some other time, human activity and interaction are so of diversity that it is not enough or even it is not neccessary to follow a cost-benefit logic, especially when you have hard time to identify who are beneficiaries and who are victims.

Here is a word I want to share with everyone: we can only and will only win the world by love and responsibility, not by proving; because essentially everything can be proved while nothing cannot be proved eventually.
                                              -- Haoying Wang, 2010

The Nature of Agricultural Economics (1)

The nature, foundation, structure and future of agricultural economics has been of concern for a long time. Even though after 90's it is noticed that agricultural economics and its education have been experiencing a downturn, it is still holding the frontier of applied econometrics and environmental economics, which are something heading future. When we look back twenty years, where lots of concerns and thoughts stacked from.

Agricultural Economics is Applied Agricultural economics is by its very nature an applied discipline-a discipline that focuses on the application of economic rinciples taken from general economics to practical, applied problems based on keen observation of the behavior of individuals, groups and institutions within an economic setting. Some agricultural economists argue that despite its reliance on economic theory, nearly all the research being conducted by agricultural economists is applied - in that the research has as its core basis observable economic phenomena based upon human behavior. Like theoretical physics related to the origins of the universe, much of the most advanced economic research being conducted in what are regarded as the best economics graduate schools has little grounding in observable economic phenomena, and consists of abstract mathematical proofs of economic theories that are seldom verifiable based on data gathered from the real world.
                                                        --David L. Debertin, 1999.


There is decreasing diversity among economics departments with respect to what is taught among the top-ten schools - that because of the inter-hiring only within the small group of schools thought to be in the peer group, there is little diversity in what is taught or in methodological approaches to research considered acceptable. As I look at the agricultural economics top-ten list, however, I see considerably greater diversity in the kinds of graduate education that would be obtained. An agricultural economics Ph.D. from Purdue would be very different from one obtained from UC-Berkeley, and no one would characterize a North Carolina State ag. econ. Ph.D. as being a clone of one produced by UW-Madison! In my view-the diversity of these graduate programs, along with the additional diversity contained in lower-ranked schools--is a source of great strength in agricultural economics, not a weakness.

                                                        --David L. Debertin, 1999.

Why Generalized Least Square Estimator?

It is known that heteroskedasticity affects the properties of the OLS estimatror (though still unbiased, but less efficient, namely larger variance). When you draw a scatter plot on raw data, the higher absolute values of the residuals to the right in the graph indicate that there is a positive relationship between the error variance and the independent variable. With this kind of error pattern, a few additional large positive errors near the right in this graph would tilt (make something move, into a position with one side or end higher than the other) the OLS regression line considerably. A few additional large negative errors would tilt it in the opposite direction considerably. In repeated sampling these unusual cases would average out, leaving the OLS estimator unbiased, but the variation of the OLS regression line around its mean will be greater - i. e., the variance of βOLS will be greater. The Generalized Least Square (GLS) technique pays less attention to the residuals associated with high-variance observations (by assigning them a low weight in the weighted sum of squared residuals it minimizes) since these observations give a less precise indication of where the true regression line lies. This avoids these large tilts, making the variance of βGLS smaller than that of βOLS.

In the case of that Durbin-Watson test indicates autocorrelated errors. It is typically concluded that estimation via Feasible GLS is called for. This is not always appropriate, however, the significant value of the Durbin-Watson statistic could result from an omitted explanatory variable, an incorrect functional form, or a dynamic misspecification. Only if a researcher is satisfied that none of these phenomena are responsible for the significant Durbin-Watson statistic value should estimation via feasible GLS proceed.

Two Nonparametrics

In the world of econometrics, the term nonparametric basically refers to the flexible functional form of the regression curve. However, there are other notions of "nonparametric statistics" which refer mostly to distribution-free methods. In the econometric context, generally, neither the error distribution nor the functional form of the mean function is prespecified.
Between the parametric econometrics and nonparametric econometrics, the question of which approach should be taken in data analysis was a key issue in a bitter fight between Pearson and Fisher in the twenties. Fisher pointed out that the nonparametric approach gave generally poor efficiency whereas Pearson was more concerned about the specification question. Both viewpoints are interesting in their own right. Pearson pointed out that the price we have to pay for pure parametric fitting is the possibility of gross misspecification resulting in too high a model bias. On the other hand, Fisher was concerned about a too pure consideration of parameter-free models which may result in more variable estimates, especially for small sample size n.

Orthogonality in Econometrics

In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle.

In linear algebra, an orthogonal matrix is a square matrix with real entries whose columns (or rows) are orthogonal unit vectors (i.e., orthonormal). Because the columns are unit vectors in addition to being orthogonal, some people use the term orthonormal to describe such matrices.
Equivalently, a matrix Q is orthogonal if its transpose is equal to its inverse:

    alternatively,   

The concept of orthogonality tends to be very important in econometrics, since we have been building almost all of the methods and rules based on the matrix platform. For example, if it happens that a relevant independent variable is omitted, in general, the OLS estimator of the coefficients of the remaining variables is biased. If the omitted variable is orthogonal to the included variables, the slope coefficient estimator will be unbiased; the intercept estimator will retain its bias unless the mean of the observations on the omitted variable is zero.
In the case of inclusion of an irrelevant variable, unless the irrelevant variable is orthogonal to the other independent variables, the variance-covariance matrix βOLS becomes larger; the OLS estimator is not as efficient. Thus in this case the MSE of the estimator is unequivocally raised.

Borrow 500 Years of Life from the Heaven

Lyrics:     Junyi Zhang, Xiaobin Fan
Compostion: Ke Fu
Translation: Haoying Wang

♣ Along the gentle waviness of rising and subsiding territory


♣ Galloping on the beloved land, beloved plateau and Yangtze South


♣ In the face of ice blade and sword, accompanied by attaching wind and rain


♣ Being cherished of my golden life from heaven


♣ And full of fraternity all along


♣ Being afraid of nothing


♣ And full of lofty sentiments all along


♣ Life is always of half pain and half enjoyment


♣ But with distinct cut between good and evil


♣ All come true in the dream for future


♣ Clanking iron heel, Never stops on the vast beloved land


♣ Standing on the top of surge, and holding


♣ The movement of universe


♣ Praying for the world of mortals


♣ Full of peace and bliss


♣ And another 500 Years from the Heaven for me


♣ Another 500 Years from the Heaven for me

Specification Problems and Empirical Study

Peter Kennedy wrote: Econometric textbooks are mainly devoted to the exposition of econometrics for estimation and inference in the context of a given model for the data-generating process. The more important problem of specification of this model is not given much attention, for three main reasons: (1) specification is not easy; (2) most of econometricians would agree that specification is an innovative/imaginative process that cannot be taught; (3) there is no accepted "best" way of going about finding a correct specification. (Of course, this is why we can always contribute something here, it is too hard to find a best and perfect way of specification.)

So the issue can come as how much trust do we have in econometrics, different people express in a different way:
All models are wrong, but some are useful. - George Box
Models are to be used, but not to be believed. -Theil, H.


Here is what Edward E. Leamer contributed into the discussion:
When an inference is suspected to depend crucially on a doubtful assumption, two kinds of actions can be taken to alleviate the consequent doubt about the inferences. Both require a list of alternative assumptions. The first approach is statistical estimation which uses the data to select from the list of alternative assumptions and then makes suitable adjustments to the inferences to allow for doubt about the assumptions. The second approach is a sensitivity analysis that uses the alternative assumptions one at a time, thereby demonstrating either that all the alternatives lead to essentially the asame inferences or that minor changes in the assumptions make major changes in the inferences. For example, a doubtful variable can simply be included in the equation (estimation), or two different equations can be estimated, one with and one without the doubtful variable (sensitivity analysis).
Simplification is a third. The intent of simplification is to find a simple model that works well for a class of decisions. A specification search can be used for simplification,as well as for estimation and sensitivity analysis. the very prevalent confusion among these three kinds of searches ought to be eliminated since the rules for a search and measures of its success will properly depend on its intent.

Again, Peter Kennedy gave following summarization: 
♣ Models whose residuals do not test as insignificantly different from white noise (random errors) should be initially viewed as containing a misspecification, not as needing a special estimation procedure.
♣ "Testing down" is more suitable than "Testing up"; one should begin with a general, unrestricted model and then systematically simplify it in light of the sample evidence.
♣ Tests of misspecification are better undertaken by testing simultaneously for several misspecifications rather than testing one-by-one for these misspcifications.

Likelihood Ratio, Wald, Lagrange Multiplier Tests

The F test is applicable whenever we are testing linear restrictions in the classic normal linear regression model. However, if, (1) the restrictions are nonlinear; (2) the model is nonlinear in the parameters; (3) the errors are distributed non-normally; then we need other asymptotically equivalent tests.

Suppose the restriction being tested is written as g(β), satisfied at the value βMLE-R where the function g(β) cuts the horizontal axis (please refer to the graph at the bottom). Then we have three asymptotically equivalent tests available to do the test and make reference, all of them are distributed asymptotically as chi-square with degrees of freedom equal to the number of restrictions being tested.

(1) The Likelihood Ratio Test: if the restrictions is true, then ln(LR), the maximized value of ln(L) imposing the restrictions, should not be significantly less than ln(Lmax), then unrestricted maximum value of ln(L). The Likelihood Ratio test tests whether [ln(LR)-ln(Lmax)] is significantly different from zero.

(2) Wald Test: if the restriction g(β)=0 is true, then g(βMLE) should not be significantly different from zero. The Wald test tests whether βMLE (the unrestricted estimate of β) violates the restriction by a significant amount.

(3) Lagrange Multiplier Test: The log-likelihood function of ln(L) is maximized at point A where the slope of ln(L) with respect to β is zero. If the restriction is true, then the slope of ln(L) at point B should be significantly different from zero. The Lagrange Multiplier test tests whether the slope of ln(L), evaluated at the restricted estimate, is significantly different from zero.

Graph for reference:

Future and Complexity

--For the Understanding of Environmental Economics and Studies Concerned

I believe that man has the power, the intelligence, and the imagination to extricate himself from the serious predicament that now confronts him. The necessary first step toward wise action in the future is to obtain an understanding of the problems that exist. This in turn necessitates an understanding of the relationships between man, his natural environment, and his technology.

                                                            -Ocho Rios, Jamaica, April 1953.

In principle, the vast knowledge we have accumulated during the last 150 years makes it possible for us to look into the future with considerably more accuracy than could Malthus. But in actual fact we are dealing with an extremely complex problem which cuts across all of our major fields of inquiry and which, because of this, is difficult to unravel (to explain something that is difficult to understand or is mysterious) in all of its interlocking aspects. The complexity of the problem, our confusion, and our prejudices, have combined to form a dense fog that has obscured the most important features of the problem from our view - a fog which is in certain respects even more dense than that which existed in Malthus’ time. As a result, the basic factors that are determining the future are not generally known or appreciated.

In spite of the complexity of the problem which confronts us, its overwhelming importance, both to ourselves and to our descendants, warrants our dissecting it as objectively as possible. In doing so we must put aside our hatreds, desires, and prejudices, and look calmly upon the past and present. If we are successful in lifting ourselves from the morass (an unpleasant and complicated situation that is difficult to escape from) of irrelevant fact and opinion and in divorcing ourselves from our preconceived ideas, we will be able to see mankind both in perspective and in relation to his environment. In turn we will be able to appreciate something of the fundamental physical limitations to man’s future development and of the hazards which will confront him in the years and centuries ahead.

Rejection From Yale

2/23/2010

Dear Mr. Wang:

Thank you very much for applying to the Graduate School of Arts and Sciences at Yale University. I regret to inform you that we are unable to offer you admission. As you know, the very high number of extraordinary candidates among our 10,400 applicants far exceeds the number of places we have in each program, and we are not able to admit many excellent candidates.

We are using this system of electronic notification to communicate with you five to ten days more rapidly than we could by letter and, therefore, help applicants plan their futures quickly and effectively. We wish you every success in all your endeavors.

Sincerely,

Jon Butler
Dean of the Graduate School

Why Student's T-test? (Part 2)

An approximate answer to the right question is worth a great deal more than a precise answer to the wrong question. 

--The first golden rule of mathematics, sometimes attributed to John Tukey

With many calculations, one can win; with few one cannot. How much less chance of victory has one who makes none at all! 

--Sun Tzu 'Art of War'

The T-test may be used to compare the means of a criterion variable for two independent samples or for two dependent samples (ex., before-after studies, matched-pairs studies), or between a sample mean and a known mean (one-sample t-test). In regression analysis, A T-test can be used to test any single linear constraint. Nonlinear constraints are usually tested by using a W, LR or LM test, but sometimes an "asymptotic" T-test is encountered: the nonlinear constraint is written with its right-hand side equal to zero, the left-hand side is estimated and then divided by the square root of an estimate of its asymptotic variance to produce the asymptotic T statistics.

For example, here is the formula to test mean difference for the case of equal sample sizes, n, in both groups:

Let E be the experimental condition and let C be the control condition. Let m be the means, s the standard deviations, and n be the sample size. Then
t = (m_E - m_C) / SQRT[(s²_E + s²_C) / n ] 

Three Different Types of T-test:

(1) One-sample T-tests test whether the mean of one variable differs from a constant (ex., does the mean grade of 72 for a sample of students differ significantly from the passing grade of 70?). When p<.05 the researcher concludes the group mean is significantly different from the constant.

(2) Independent sample T-tests are used to compare the means of two independently sampled groups (ex., do those working in high noise differ on a performance variable compared to those working in low noise, where individuals are randomly assigned to the high-noise or low-noise groups?) . When p<.05 the researcher concludes the two groups are significantly different in their means. This test is often used to compare the means of two groups in the same sample (ex., men vs. women) even though individuals are not (in the case of gender, cannot be) assigned randomly to the two groups (to "men" and to "women"). Random assignment would have controlled for unmeasured variables. This opens up the possibility that other variables either mask or enhance any apparent significant difference in means. That is, the independent sample t-test tests the uncontrolled difference in means between two groups If a significant difference is found, it may be due not just to gender; control variables may be at work. The researcher will wish to introduce control variables, as in any multivariate analysis. 

(3) Paired sample T-tests compare means where the two groups are correlated, as in before-after, repeated measures, matched-pairs, or case-control studies (ex., mean candidate evaluations before and after hearing a speech by the candidate). The algorithm applied to the data is different from the independent sample t-test, but interpretation of output is otherwise the same.

Associated Assumptions:

(1) Approximately Normal Distribution of the measure in the two groups is assumed. There are tests for normality. The t-test may be unreliable when the two samples come from widely different shaped distributions (see Gardner, 1975). Moore (1995) suggests data for t-tests should be normally distributed for sample size less than 15, and should be approximately notmal and without outliers for samples between 15 and 40; but may markedly skewed when sample size is greater than 40. 

(2) Roughly Similar Variances: There is a test for homogeneity of variance, also called a test of homoscedasticity. In SPSS homogeneity of variances is tested by "Levene's Test for Equality of Variances", with F value and corresponding significance. There are also other tests for homogeneity of variances. The T-test may be unreliable when the two samples are unequal in size and also have unequal variances (see Gardner, 1975). 

(3) Dependent/Independent Samples. The samples may be independent or dependent (ex., before-after, matched pairs). However, the calculation of T differs accordingly. In the one-sample test, it is assumed that the observations are independent. 

One last note is that, don't confuse a T test with analyses of a contingency table (Fishers or chi-square test). Use a T test to compare a continuous variable (e.g., blood pressure or weight). Use a contingency table to compare a categorical variable (e.g., pass vs. fail, viable vs. not viable). 

Reference:

Gardner, P. L. (1975). Scales and statistics. Review of Educational Research. 45: 43-57. Discusses assumptions of the t-test. 

Moore, D. S. (1995). The Basic Practice of Statistics. NY: Freeman and Co. 

Why Student's T-test? (Part 1)

Here I am trying to answer two questions for myself:

1. What is the difference between Z-test and T-test?
2. Why we need student's T-test?

First, let's be clear on Z-test V.S. T-test. A thumb rule can be referred as, Z-test is used when the sample size is more than 30 while T-test is used for smaple size less than 30. Now let's get back to the history of story:

Sometimes, measuring every single piece of item is just not practical. That is why we developed and use statistical methods to solve problems. The most practical way to do it is to measure just a sample of the population. Some methods test hypothesis by comparison. The two of the more known statistical hypothesis tests are the T-test and the Z-test. Let's try to break down the two.

Strictly speaking, the Z-test is a test for populations rather than samples. In the real world though, either test will give you a pretty close answer. using the T-test is more accurate because the sample deviation is specific and tailored to the sample you are studying, so the answer will be more accurate. When using a T-test of significance, it is assumed that the observations come from a population which follows a Normal distribution. This is often true for data that is influenced by random fluctuations in environmental conditions or random measurement errors. Whereas the T-distribution is essentially a corrected version of the normal distribution in which the population variance is unknown and hence is estimated by the sample standard deviation.

There are various T-tests and two most commonly applied tests are the one-sample and paired-sample T-tests. One-sample T-tests are used to compare a sample mean with the known population mean. Two-sample T-tests, the other hand, are used to compare either independent samples or dependent samples.

As mentioned above, T-test is best applied, at least in theory, if you have a limited sample size (n < 30) as long as the variables are approximately normally distributed and the variation of values in the two groups is not reliably different. It is also great if you do not know the populations’ standard deviation. If the standard deviation is known, then, it would be best to use another type of statistical test, the Z-test. The Z-test is also applied to compare sample and population means to know if there’s a significant difference between them. Z-tests always use normal distribution and also ideally applied if the standard deviation is known. Z-tests are often applied if the certain conditions are met; otherwise, other statistical tests like T-tests are applied in substitute. Z-tests are often applied in large samples (n > 30). When T-test is used in large samples, the T-test becomes very similar to the Z-test. There are fluctuations that may occur in T-tests sample variances that do not exist in Z-tests. Because of this, there are differences in both test results.

Summary:


1. Z-test is a statistical hypothesis test that follows a normal distribution while T-test follows a Student’s T-distribution.
2. A T-test is appropriate when you are handling small samples (n < 30) while a Z-test is appropriate when you are handling moderate to large samples (n > 30).
3. T-test is more adaptable than Z-test since Z-test will often require certain conditions to be reliable. Additionally, T-test has many methods that will suit any need.
4. T-tests are more commonly used than Z-tests.
5. Z-tests are preferred than T-tests when population standard deviations are known.

Maybe We Just Need a New Word: Gadget

"Samsung has just announced at Barcelona a new cell phone, the Beam, that they expect to have on the market this summer. Its special feature is a built-in pico projector, making it a combination cell phone and (very wimpy) video projector. A cute gadget, although not one that I am likely to have much use for. I do, however, have one suggestion for improving it."

Reading through this news, I am happened to be interested in the word Gadget: Two similar explanations can be easily referenced from dictionary:

(1) an often small mechanical or electronic device with a practical use but often thought of as a novelty;

(2) any object that is interesting for its ingenuity or novelty rather than for its practical use.

So it comes to me as a question, have we been proposing and digging Gadgets in econometrics and economics? This could happen to be a 'gadget' question, but it is definitely not a 'gadget' issue. Too many people are publishing papers which probably are going to have its author(s) as the only and last careful reader. So why we spend one or two years, even three years to invent such a "gadget"? For tenure, for promotion or just for fun (self understanding of the subjects)? Maybe it is just for a popular social demand of vanity, maybe it is just an indispensable part of the system, who knows?