# Questions tagged [method-of-moments]

A method of parameter estimation by equating sample and population moments then solving the equations for the unknown parameters.

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### Method of Moments of Uniform Distribution

Let $x_1=2, x_2 = 1, x_3 = \sqrt5, x_4 = \sqrt2$ be the observed values of a random sample of size 4 from a uniform distribution $U(-\theta, \theta)$ where $\theta>0$. Then the method of moments ...
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### Why can the method of moments be expressed as a minimization problem?

Generalized method of moments (GMM) estimation seems to be called generalized method of moments because the standard method of moments (MoM) is a special case, following the following logic. MoM is ...
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### Why is method of moments (MoM) not unique? What is uniqueness?

I just learned that a MoM estimator is not unique... What does that mean? What does being unique mean in general? For example, if we were to find the mean using a MoM estimator for a uniform ...
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### OLS and the Method of Moments estimator

I have been taught that the Method of Moments Approach could be used to obtain estimates of a bivariate regression of the form y = B0 + B1x + u. Specifically, the 2 Moment conditions are E(u) = 0 and ...
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### Population and sample moment notation

A population moment is often stated as $E[x]$ the sample moment counterpart is stated as $1/n \sum_{i=1}^{n} x_i$. I do not quite understand what is meant by $E[x]$. Why not $E[x_i]$? What does $E[x]$ ...
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### Is a correction needed in the weighting matrix when using the simulated method of moments?

I am following several guides on the simulated method of moments (SMM) and one practical way of getting the weighting matrix, W, is to bootstrap the empirical moments and invert their covariance ...
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### Matching Simulated Moments Perfectly in Practice

Several sources suggest that when estimating a model using the simulated method of moments (SMM), one ought to always be able to get the difference between the empirical and simulated moments to be 0 ...
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### Why is OLS related to Moment Estimation?

I am trying to understand the relationship between OLS and the Method of Moments. Specifically, why OLS is considered a special case of Method of Moments ... and why Method of Moments is a special ...
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### Estimating the population parameters using the sample moments

I have distribution , which is not normal and highly skewed. I take a sample from that distribution and calculate the first four moments of the sample distribution. The first four moments calculated ...
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### For Gamma distribution, use MLE or MoM?

For Gamma distribution, is it better to use MLE(maximum likelihood estimation) than MoM(method of moments) to estimate the shape and scale parameters? Also, in python SciPy, does gamma.fit use MLE? I'...
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### Estimating $1/a$ for following pdf using method of moments estimation

A random sample of size $n$ is being drawn from a population with pdf as: $$f(x) = \begin{cases} (a + 1)x^a & \text{for }0<x<1, \\ 0 & \text{otherwise.} \end{cases}$$ Can we express the ...
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### Using the methods of moments in R for the dirichlet distribution

I'm trying to build a distribution of transition probabilities to randomly sample from in a Markov model where individuals can transition from one health state to another (assume that in the image ...
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The Pareto distribution has the following $cumulative \ distribution \ function$ : $$F(x;\alpha ,\Theta ) = \left\{\begin{matrix} 1 - (\frac{\alpha}{x})^{\theta}\ \ if \ \alpha \leq x\ & \\ 0 \ ... • 423 4 votes 2 answers 362 views ### Finding method of moments estimate for density function f(x|\alpha) = \frac {\Gamma(2\alpha)} {\Gamma(\alpha)^2}[x(1-x)]^{\alpha - 1} Suppose that X_1, X_2, ..., X_n are i.i.d random variables on the interval [0,1] with the density function$$ f(x|\alpha) = \frac {\Gamma(2\alpha)} {\Gamma(\alpha)^2}[x(1-x)]^{\alpha - 1} $$where ... • 275 2 votes 1 answer 557 views ### Efficiency of IV vs GMM I am trying to understand how IV/just identified GMM and overidentified GMM compare when it comes to efficiency. The way I understand it, we are able to identify the vector of coefficients in IV and ... • 41 4 votes 2 answers 207 views ### Method of moments for symmetric mean zero distributions When using the method of moments to fit a symmetric mean zero distribution, does it make more sense to fit higher order moments or lower order absolute moments? I could not find any resources which ... 4 votes 3 answers 285 views ### What's special about moments that allows "method of moments" to work? The idea behind Method of Moments (MOM) is quite intuitive: find the parameter values so that the population moments (which are functions of those parameters of interest) matches the sample moments. ... • 4,600 2 votes 1 answer 217 views ### Consistent but inefficient GMM Consider the following linear model$$y_t = x_t' \beta +u_t$$where t =1,...,T and x_t = (x_{1t} x_{2t} ... x_{kt})' ,  \beta is k \times 1 vector of unknown coefficients, u_t is an iid ... • 1,140 1 vote 0 answers 66 views ### Discuss that method of moment estimation is inefficient Discuss that method of moment estimation is inefficient. Then model is Consider the following linear model$$y_t = x_t' \beta +u_twhere t =1,...,T and x_t = (x_{1t} x_{2t} ... x_{kt})' ,  \... • 1,140 0 votes 0 answers 331 views ### When is the Optimal weighting matrix in GMM singular? currently I am trying to estimate a simple linear regression: $$y_t = X \beta + \varepsilon_t,$$ where I try to find 4 coefficients and one specific predictor is an ... 2 votes 1 answer 85 views ### How does the information in the problem statement and this solution align with the provided description of the method of moments? I have the following problem: Let Y_1, Y_2, \dots, Y_n be i.i.d. \text{Uniform}(\theta, 1) random variables, and let an estimator be \hat{\theta} = \min\{ Y_1, Y_2, \dots, Y_n \}. You may find ... • 2,096 3 votes 1 answer 563 views ### Negative-Binomial Method of moments with an offset Given the method-of-moments approach to estimate the parameters of the NB-2 distribution \mu and \phi: \mu = \bar{y}  \phi = \frac{\bar{y}^2}{s^2 -\bar{y}} $$How can this be extended to ... • 53 0 votes 1 answer 127 views ### Method of Moment for Gamma Distribution I know that the empirical r-th moment is defined as:$$\hat E(X^r) = \frac{1}{n} \sum_{i=1}^n x_i^r $$So for the first moment I did:$$E_{\lambda,\alpha}(X) = \hat E(X) = \bar X \bar X = \...
The statement I am given the following discrete distribution with $\theta>0$ $$p(x) = \left(\frac{\theta}{1+\theta}\right) ^{2-x}\left(\frac{1}{1+\theta}\right)^{x-1} \hspace{1cm} x=1,2$$ I need to ...
Suppose a random variable follows the logistic distribution, $X ∼ Logistic (\mu, \sigma)$ and we restrict our attention to random samples drawn from this random variable $X$. What would be the MoM and ...