The answer is no. The model contains the value for X3 and if your newdata does not contain this value it will throw an error.
See following example. I set mtcars in a dataframe, remove the Y column and one of the X columns (cyl). Create a simple lm model based on mtcars and predict on df.
df <- mtcars
df$mpg <- NULL # remove Y
df$cyl <- NULL # remove one X value
lm_model <- lm(mpg ~ ., data = mtcars)
predict(lm_model, df)
Error in eval(expr, envir, enclos) : object 'cyl' not found
What you could do is impute the missing column to your newdata with the value 0, mean, median or use an imputation function. But then you should realize that your model predictions will be off.
Edit based on comment:
Off course, you can skip the X3 column and just regress on X1 and X3. Imputing the value is always a decision based on experience. Median in case of outliers, mean if the data is normally distributed, or imputing if you can deduce the value of X3 based on values of X1 and X2. For example, you could look into the package mice and it's vignette. There are more packages for imputing missing data, but it some of it depends on why the data is missing in the first place. There is a whole body of work on how to deal with missing data.
Here is a link to chapter 25 of Andrew Gelman's book Data Analysis Using Regression and Multilevel/Hierarchical Models dealing with missing data.
and here is a link to a list of software package dealing with missing data. The author of mice also wrote an article on this package in the journal of statistical software which you can find here
Ordinarily, we interpret coefficients in terms of how the expected value of the response should change when we effect tiny changes in the underlying variables. This is done by differentiating the formula, which is
$$E\left[\log Y\right] = \beta_0 + \beta_1 x_1 + \beta_2\left(\frac{x_3}{x_1}\right).$$
The derivatives are
$$\frac{\partial}{\partial x_1} E\left[\log Y \right] = \beta_1 - \beta_2\left( \frac{x_3}{x_1^2}\right)$$
and
$$\frac{\partial}{\partial x_3} E\left[\log Y \right] = \beta_2 \left(\frac{1}{x_1}\right).$$
Because the results depend on the values of the variables, there is no universal interpretation of the coefficients: their effects depend on the values of the variables.
Often we will examine these rates of change when the variables are set to average values (and, when the model is estimated from data, we use the parameter estimates as surrogates for the parameters themselves). For instance, suppose the mean value of $x_1$ in the dataset is $2$ and the mean value of $x_3$ is $4.$ Then a small change of size $\mathrm{d}x_1$ in $x_1$ is associated with a change of size
$$\left(\frac{\partial}{\partial x_1} E\left[\log Y \right] \right)\mathrm{d}x_1 = (\beta_1 - \beta_2(4/2^2))\mathrm{d}x_1 = (\beta_1 - \beta_2)\mathrm{d}x_1.$$
Similarly, changing $x_3$ to $x_3+\mathrm{d}x_3$ is associated with change of size
$$\left(\frac{\partial}{\partial x_3} E\left[\log Y \right] \right)\mathrm{d}x_3 = \left(\frac{\beta_{2}}{2}\right)\mathrm{d}x_3$$
in $E\left[\log y\right].$
For more examples of these kinds of calculations and interpretations, and to see how the calculations can (often) be performed without knowing any Calculus, visit How to interpret coefficients of angular terms in a regression model?, How do I interpret the coefficients of a log-linear regression with quadratic terms?, Linear and quadratic term interpretation in regression analysis, and How to interpret log-log regression coefficients for other than 1 or 10 percent change?.
Best Answer
Aksakal's answer is correct. By controlling for all variables in a regression, you "keep them constant" and you are able to identify the partial correlation between your regressor of interest. Let me give you an example to make this clearer.
First, let us create some correlated $X$s.
Now, since all these variables are generated by some underlying variable $ex$, they are clearly correlated. You can check this using
cor(x1,x2), for instance.Now, let us generate the dependent variable with known parameters.
Here we know that $\beta_1=1, \beta_2=2, \beta_3=3$. I have picked them arbitrarily. Let us now see if Aksakal's approach can uncover these parameters:
If it works, the estimated parameters should be close to the ones we have picked. Here the result:
As you can see, all parameters have been uncovered.
Having said that, there are many caveats involved here as well. Most importantly, you should not interpret these coefficients in a causal way. Depending on your actual situation, it might help if you explain a bit more what you are trying to estimate so that people can evaluate whether this method is appropriate (or whether answering your research question is feasible at all). For instance, why do you think your independent variables are correlated? Is it that $X_1$ might have an effect on $X_2$ and this has an effect on $y$? If this is the setup you have in mind, then depending on your field, you may want to look into mediator/moderator analysis or into quasi-experimental methods. Hence you see you might benefit from elaborating a bit more on your situation.