7 votes
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Regression Optimization problem under constraints

The situation is given in the following picture The black line is the true conditional mean $E(y|x)$. If we truncate the data, all observations above the truncation $Y^A$ are not observed. For low ...
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  • 8,642
7 votes
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Econ Intuition for Jacobian inverse in demand system

For the 2x2 case being considered, write $$\mathbf{B}=\left[\begin{array}{cc} b_{1,1} & b_{1,2}\\ b_{2,1} & b_{2,2} \end{array}\right].\quad$$ It follows that the element (1,1) in $B^{-1}$ is ...
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  • 595
5 votes
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Budget hyperplane in n dimensions

Let matrix $A = \begin{bmatrix} p_1 & p_2 & \ldots & p_n \end{bmatrix}$. Let $\mathbf{x}^*$ be a fixed solution to $A \mathbf{x} = c$. Then for any vector $\mathbf{u}$ that belongs to the ...
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3 votes

Calculating natural rate of unemployment

The model you fit is simply inadequate to estimate the natural rate of unemployment any results from it will be completely unreliable, so I am not surprised if they make no sense. Furthermore, natural ...
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  • 43.5k
3 votes
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Why the definition of productive economy in Leontief Open Model is such?

This is a specific terminology employed in the literature on Leontief model. Productive here means that all sectors must be profitable (do not confuse it with notions of productivity used elsewhere in ...
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  • 43.5k
3 votes

How do you convert or move from a linear cost function to a quadratic cost function?

Your eq (2.10) is not more general than (2.9), but corresponds to an alternative specification. A more general version would be: $$ C_i(Q_i)=FC_i+a_{1,i}Q_i+a_{2,i}Q^2_i. $$ This specification allows ...
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  • 2,692
3 votes
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When gradient of utility function is a zero vector

This concerns the partial derivatives of the utility function with respect to goods, and not the partial derivative of the Lagrangian of the maximization problem. So a zero derivative, and moreover ...
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2 votes

How to deal with a singular Leontiev inverted matrix?

I am not sure exactly what you divided by what, but suppose your input-output table looks like this: $$ A = \left[ \begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{...
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  • 26.7k
2 votes

How can I write a conditional expectation of finite state markov process in matrix notation

Let the $n$ states of the finite-state markov chain be denoted by $\{x_1,...,x_n\}$ and let $\vec e = [e(x_1), ..., e(x_n)]'$. Now, first note that because $X_{t+1} \mid X_t$ is independent of $W_{t+1}...
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  • 9,155
2 votes
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How to utilize the projection matrix in econometrics?

This kind of projections in econometrics are usually employed for partialling out some covariates from a linear regression. Observe that in general $P_X\neq I$. Consider $X=\begin{bmatrix}1&5\\1&...
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  • 342
2 votes

Derive the growth rate of an equation

First, note that the growth rate of $\mu$ is defined as $\dot\mu = \frac{ d\mu }{ \mu }$. Therefore, you will have to take the total differential of the equation; and divide by $\mu$. For the first ...
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  • 263
2 votes

Hep with total differentiation of an AD function

For a variable $X$, let $dX$ denote its total differential. Let $k$ be a constant, and $X$ and $Y$ variables. You'll need the following rules: $$dk = 0$$ (constant rule), $$d(X + Y) = dX + dY$$ (...
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  • 263
2 votes
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Negative Definite vs Semi-definite Hessian - Sufficient vs Necessary conditions?

The simplest example is $-x^3$ in the single variable case, or $-x_1^3-x_2^3$ in the case of two variables. The Hessian matrix is negative semi-definite at $(0,0)$, but there is no maximum at this ...
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  • 2,692
2 votes

Understanding utility function curve and marginal rate of substitution

This is an expansion on @1muflon1's answer. Motivation and Applications In the utility function, $\alpha$ allows for an alternative to a linear growth function, especially when the parameters are ...
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1 vote
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What are the concepts in Linear Algebra that model the idea of Identification Strategy in Econometrics?

In econometrics, identification can take several forms depending on the type of model you are working with (see this survey for a more comprehensive description). However, in general, identification ...
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  • 4,158
1 vote
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$x\sim y$ implies $x+a\sim y+a$ for any $a\geq0$ and $x,y\in\mathbb R^n$, then the preference is linear?

It is not true. Let us consider $\mathbb{R}^2$ so bundles are $x = (x_1,x_2)$. Consider the preference: (i) If $x_1 \leq 0$, preferences are lexicographic, i.e. $$ x \succ y \Leftrightarrow \begin{...
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1 vote

How can difference equations with an infinite summation be represented in matrix form?

Maybe the differencing approach should work for you. Basically the idea is to reduce this series to a finite expression, using the lag operator. Let me explain with an example from Costa (2016, p.81) (...
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  • 717
1 vote

Leontief input output model with column sum greater than 1

You are right when saying that mathematically in the cited theorem the condition of column sums being less than 1 is not an "if and only if" condition and thus exceptional circumstances are ...
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1 vote
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Leontief input output model with column sum greater than 1

In terms of if and only if statements according to Peterson & Olinick (1982); A substochastic matrix A is productive if and only if $I-A$ is nonsingular. In substochastic matrix the sum of ...
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  • 43.5k
1 vote

Calculating the elasticity of substitution between factors of production

First of all, I think that 'linear and homogeneous' is a typo of 'linearly homogeneous.' Indeed, it can be shown that if the production function $V$ is linearly homogeneous, Allen Elasticity of ...
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  • 66
1 vote

Has this differential calculus inequality approach to optimizing the production possibility curve exist?

I don't really understand your optimization problem as it stands. Just as a few examples, when you write out $$\min \sum_m \sum_v \left( \frac{\partial \vec f_v (\vec Q_v)}{\partial \vec f_m (\vec ...
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  • 6,409
1 vote

Budget hyperplane in n dimensions

I got some outside help for the ending of the proof I was attempting. I'll leave this question if by chance someone else finds it useful. So if we want to show $p_1x_1 + \cdots + p_{n−1}x_{n−1} = 0 \...
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  • 6,409
1 vote

How to deal with a singular Leontiev inverted matrix?

All right kids. You gave me the answer when Dismalscience wrote "this is still not a problem as long as the value of the good produced differs from the sum of the value of its inputs" ... actually, ...
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1 vote

How to deal with a singular Leontiev inverted matrix?

I didn't divided each column by the sum of the column. This would make no sence since the goal is to produce a technical coefficient matrix that links each input (row) needed by the industry (column) ...
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