I am using First difference transformation, where my variables are represented in a logarithmic form. How do I interpret the coefficients from such regressions?   Thank you in advance!
123151
My problem is interpreting coefficients of such time series model:  \begin{equation} \ln Y_t - \ln Y_{t-1} =b_1 \cdot \left(X_{t}-X_{t-1}\right)+b_2 \cdot Z_t.\end{equation}  I don't know how to interpret coefficients $b_1$ and $b_2$.  Hope that someone can help.
39766
I'm using fourth root (1/4) power transformation on my response variable, as a result of heteroscedasticity. But now I'm not sure how to interpret my regression coefficients.   I assume that I would need to take the coefficients to the fourth power when I back-transform (see below regression output). All of the variables are in units of dollar in millions, but I would like to know the change in dollar in billions.    While holding the other independent variable constant, a billion dollar change in fees, on average, leads to a change of 32 (or 32,000 dollars) in collections. I take 0.000075223 * 1000 (to get to billions) ^ 4 = 0.000032. Now do I multiply this number by 1 million or 1 billion (the original unit of the dependent variable is in millions)?  lm(formula = (Collections^(1/4)) ~ Fees + DIR)                   Estimate      Std. Error  t value            Pr(&gt;|t|) (Intercept)   2.094573355     0.112292375   18.653  0.0000000000000151 Fees        **0.000075223   **0.000008411    8.943  0.0000000131878713 DIR           0.000022279     0.000004107    5.425  0.0000221138881913
57576
If I have a linear regression of the form:  $$y \sim 1+\beta_1x_1 + \beta_2x_1x_2,$$  where $x_2$ is a Boolean variable depending on another variable $x_{2cont}$, a positive variable:  $$x_2 = \begin{cases} 1 &amp; \mbox{if } x_{2cont} &gt; 5 \\ 0 &amp; \mbox{if } 0 &lt; x_{2cont} \le 5. \end{cases}$$  And I got regression results. How shall I interpret the regression results?  esp. how shall I interpret the two coefficients I get from this regression,  $\beta_1 \text{ and } \beta_2$?
824
I have multiple regression with, say 3 independent variables: $Y=B_0+B_1x_1+B_2x_2+B_3x_3$ I would like to test if $B_2+3B_3$ is significantly different from zero, i.e.  $$H_0: B_2+3B_3=0$$ $$H_1: B_2+3B_3\neq 0$$ Can you please help to find appropriate way to test for significance of linear functions of two coefficients as in above example.  Many thanks in advance.
108977
I am trying to determine the most appropriate degree of difference to start building an arima model for this dataset. I already know that I need a first difference, but there is a seasonal component, and I am not exactly sure where to go from here. I've tried several potential options but I am just not very confident when interpreting these plots. I will include the raw time series plot / acf / pacf plots as well as the first differenced time series / acf / pacf plots. Any help would be fantastic!! Original: Time series:  acf:  pacf:  First Difference:  First diff acf:  First diff pacf:
103152
Assume a simple regression model, $y = \beta_0 + x\beta_1 + u$. I decide to change the units of measurement for the explanatory variable and the response variable. Do the $\beta_0$ and $\beta_1$ parameters change as well?  I assume that the $\beta$ parameters will stay constant.  Does anyone know for sure?  How does it change the value of $R^2$?
77293
I realize that I lack any numerical intuition for logarithms. For example, when comparing two logarithms like $\log_78$ and $\log_89$, I have to use the change-of-base formula and calculate the values with a calculator in order to determine which of the two is larger.  Can anyone demonstrate an algebraic/analytic method to find out which of the two is larger without using a calculator?
52575
I'm trying to make sense of a derivation I'm following from the lecture notes of Stanford's ML course.  Specifically the notes are here:   I'm particular I'm interested in the step where they are optimizing the lower bound of the log-likelihood by finding the derivative of that function w.r.t the mean (top of page 7). I'm a bit stuck on going from line 1 to line 2 in that sequence.  To copy here (using abbreviated notation for space):  $$ \nabla \mu_s \sum_{i=1}^{m}\sum_{j=1}^{k} w_j^{(i)}\log\frac{\alpha_j \exp\psi_{j}^{(i)}\phi_j}{w_j^{(i)}}  \qquad(1) $$ $$ \nabla \mu_s \sum_{i=1}^{m}\sum_{j=1}^{k} w_j^{(i)}\psi_{j}^{(i)}\qquad(2) $$ where $\psi_{j}^{(i)} =-\frac{1}{2}(x^{(i)}-\mu_j)^T\Sigma^{-1}_{j}(x^{(i)}-\mu_j)$ and $\alpha_j=\frac{1}{(2\pi)^{n/2}|\Sigma_j|^{1/2}}$.  The quantity $\mu_s$ is the mean (of a multivariate gaussian distribution) for class $s$.    In short, I'm not sure how they get from (1) to (2).  From just changing the log of (1) I see:  $$ \nabla \mu_s \sum_{i=1}^{m}\sum_{j=1}^{k} w_j^{(i)}\log \left(\alpha_j \exp\psi_{j}^{(i)}\phi_j\right) - w_j^{(i)}\log w_j^{(i)} \qquad(1) $$  To get to their form (2) my impression is that they take pre-emptively discard terms that would contain derivatives of $w_j^{(i)}$ (w.r.t. $\mu_s$).  At least if you assume that much, you get to (2).  However, as they show earlier in that PDF,  $$ w_j^{(i)} = p(z^{(i)}=j | x^{(i)}) =\frac{p(x^{(i)}|z^{(i)}=j)p(z^{(i)}=j) }{\sum_j p(x^{(i)},z^{(i)}=j) } $$ i.e. this quantity is the posterior distribution of the class labels.  Intuitively, it seems like $\partial w_j^{(i)}/ \partial \mu_s$ should be non-zero.  After all, changing the location of the mean $\mu_s$ would change your posterior probability for a particular class given the same data.  (Right?).  Just from a mechanical point of view, $\partial w_j^{(i)}/\partial \mu_j$ would seemingly have some non-zero terms since $p(x^{(i)}|z^{(i)}=j) \sim \mathcal{N}(\mu_j, \Sigma_j)$.  If I work through the algebra, I get  $$ \frac{\partial w_j^{(i)}}{\partial \mu_j} = w_j^{(i)}(1-w_j^{(i)})\frac{\partial \psi^{(i)}_{j}}{\partial \mu_j} $$  What am I missing here?  If I take the log-likelihood (without even thinking about EM algorithm),    $$ \ell = \sum_{i=1}^{m}\log \sum_{j=1}^{k}p(x^{(i)}|z^{(i)}=j)p(z^{(i)}=j) $$ and differentiate w.r.t $\mu_s$, I correctly get  $$ \mu_s^* = \frac{\sum_{i=1}^{m}w_s^{(i)}x^{(i)}}{\sum_{i=1}^{m}w_s^{(i)}} $$  so I'm comfortable with the eventual result, just not the intermediate steps.
51132
How can I show that? I've tried to reverse the logarithm to it's exponential form in a trial to show that but I got no success. Can you help me?
20528
I'm fitting a regression on the $\log(y)$.  Is it valid to back transform point estimates (and confidence/prediction intervals) by exponentiation?  I don't believe so, since $E[f(X)] \ne f(E[X])$ but wanted other's opinions.  My example below shows conflicts with back transforming (.239 vs .219).  set.seed(123)  a=-5 b=2  x=runif(100,0,1) y=exp(a*x+b+rnorm(100,0,.2)) # plot(x,y)  ### NLS Fit f &lt;- function(x,a,b) {exp(a*x+b)}  fit &lt;- nls(y ~ exp(a*x+b),  start = c(a=-10, b=15))  co=coef(fit) # curve(f(x=x, a=co[1], b=co[2]), add = TRUE,col=2,lwd=1.2)  predict(fit,newdata=data.frame(x=.7)) [1] 0.2393773  ### LM Fit # plot(x,log(y)) # abline(lm(log(y)~x),col=2) fit=lm(log(y)~x) temp=predict(fit,newdata=data.frame(x=.7),interval='prediction') exp(temp)         fit       lwr       upr 1 0.2199471 0.1492762 0.3240752
49240
I'm working on a VAR model and am doing a log transformation of the raw data.   x = log(x)   After differencing, running various tests and running a VAR(2) model, I add the predicted differences cumulatively to the original time series (logged) like this:  prd.diffs &lt;- as.ts(prd$mean$d.x1) U &lt;- tail(undiff,2) cumu &lt;- cumsum(prd.diffs) + U[1]   However, while the output now looks as it should, it is still in logged values.  My question is as follows:  How does one return the logged output to original values? I'm under the impression that using exp() will not give the correct results, and having tried some variations it hasn't worked for me.   Does the fact that it is a VAR($p$) model (and possibly what lag it has) influence the method, and how should that be implemented to get the correct output?  P.S. I realise that the above code does not constitute a replicable example as I didn't deem that necessary for the question, but should that be desired I can add some more code.
36588
I was reading an article about the polar motion of the Earth: . A regression is performed between observed and modelled complex time series of data - the type of regression is not specified. The regression coefficient is then calculated for different values of the estimated parameter (quality factor); its amplitude and phase are shown in Figure 2 (I don't know if I am allowed to post it directly here). My question is why can the amplitude of the regression coefficient be as high as 2.3? I thought that the absolute value of the regression coefficient, for real series, had to be in the range 0-1. Therefore I was expecting a maximum amplitude of $\sqrt{2}$ for a complex series. The definition of regression coefficient is not mentioned in the article. At first I thought it may be a linear regression since if the model is correct, the modelled and observed series should be very close; however, this seems incompatible with a value of 2.3. Is there a more general definition that I'm missing?
42347
Because my dependent variables in a multiple linear regression model are in different units, some coefficients are > 10,000 and others less than 0. I need to standardize them by using the scale function on R, but because some of my data is negative, it's impossible to take log of some of the variables which I need to do before standardizing. Is there any other step I can take? Can I standardize without log transformation first?
77317
I'm doing a Logistic Regression on company level using the total assets to control for company size. Due to the skewedness of the data, I do a log-transformation of the asset data. While I get no significance for the log-transformed variable, the not transformed variable is highly significant (p &lt; 0.01). What could be a possible explanation of such a result? Does it mean that the relationship is not diminishing for extremely high and low values or does it show an outlier problem and should not be used for interpretation of the data.
13763
I can't seem to find a good way to solve this.  I tried using L'Hopitals, but the derivative of $\log(n!)$ is really ugly. I know that the answer is 1, but I do not know why the answer is one.  Any simple way to go about this?
13365
I am doing some independent study in asymptotic statistics and point estimation and am aware that you can get from log transformations of uniform random variables (exponential) all the way up to chi-squared distributions but I came across an intriguing question and I'm not sure how to handle it. I was thinking CLT or Delta Method but am unsure of myself. The thing I am trying to sort out is:  Given $X_1,\ldots X_n\overset{iid}{\sim}Uniform(0,\theta)$ and $Z_n=\left(\prod_{i=1}^nX_i\right)^{1/n}$ what is an example of a function of $Z_n$, i.e. $g(Z_n)$, so that $g(Z_n)\overset{d}{\rightarrow}\chi_{(1)}^2$, where $\overset{d}{\longrightarrow}$ denotes convergence in distribution and $\chi_{(1)}^2$ is a chi-squared random variable with 1 degree of freedom?  Maybe something like $g(h(Z_n))$ where $h(Z_n)\overset{d}{\rightarrow}Normal(0,1)$ and $g(t)=t^2$? Any help is greatly appreciated.  I have tried:  Let $Y_1=X_i/\theta$ I get $Y\sim Uniform(0,1)$ and so $-\log(Y)\sim Exponential(1)$. From here I know $-2\log(Y)$ is $\chi_{(2)}^2$ but I don't see how to relate $-\log(Y)$ back to a function of $Z_n$ that is $\chi_{(1)}^2$
2768
I have scoured around, reading posts on Cross Validated () and also looking at references including Dobson and McCullagh and Nelder, e.g.  so I am aware that this topic is well trodden. Nevertheless, I am trying to articulate and formalize my understanding of GLM and while several posts have helped me with that, I am conscious of gaps and the possibility of an unsound foundation of my understanding.   In simple linear regression we have some set of observations $(x_i, y_i)$ pairs and treat $y_i$ as a realization of a random variable, $Y_i$, distributed as $Y_i \sim N(\mu_i, \sigma^2)$. The means ($\mu_i$) depend on the predictor but the variance is constant. We model $\mu_i = \beta_0 + (\beta_1 \times x_i)$ (or matrix equivalent ) which I believe is the same as saying $Y_i = \mu_i + \epsilon_i$ with $\epsilon_i \sim N(0,\sigma^2)$. I am not sure but I think a correct way to state $\mu$ is $\mu_i = E[Y_i|X=x]$. Can someone confirm that?  Anyway, we might transform the response to achieve linearity (maybe taking logs) and in that case we are modelling $log(Y_i) = \alpha_0 + (\alpha_1 \times x_i) + \epsilon_i$ where $\epsilon_i$ now is assumed to have a log normal distribution.   We generalize by decomposing the model into:   A structural component ($\beta_0 + (\beta_1 \times x_i)$) A link g(.) A response distribution (or random component) (member of the exponential family - Guassian, bin,gamma etc)   Let's say that the observations are assumed to come from a distribution in the exponential family and to keep things simple assume that it is the Gaussian distribution. Again, the expected value comes out as $E[Y_i|X=x] = \mu_i$, but does so from the first derivative of the b(theta) term in the exponential family form () (pg 29). In the case where we assume a binomial distribution, the expected value comes out at np but it still comes from the first derivative of the b(theta) term when the distribution is expressed in the exponential family form. Note, I don't fully appreciate or understand the derivation of the mean or variance at this stage, perhaps someone could provide a layman's explanation?  Instead of modelling the mean as was done for simple linear regression, we now model a transformation of the mean so instead of saying $\mu_i = \beta_0 + (\beta_1 \times x_i)$ we are saying $g(\mu_i) = \eta_i = \beta_0 + (\beta_1 \times x_i)$ where g is some link function (invertible and differentiable). I think this is a key (yet still somewhat confusing to me) distinction between SLR and GLM. In SLR we transform the response ($y_i$) and model that, in GLM we transform the expected value ($\mu_i$ in the case of the Gaussian example) and model that. Another way of saying it is that in the SLR case we model $E[g(Y_i|X=x)] = \beta_0 + (\beta_1 \times x_i)$ but in the GLM world we are modelling $\eta_i = g[E(Y_i|X=x)] = \beta_0 + (\beta_1 \times x_i)$.   My question is around verifying my understanding and statement of the essence of the foundations of GLM and the differences between GLM and the traditional linear model. Thanks.