How do I compute whether my linear regression has a statistically significant difference from a known theoretical line? I have some data which is fit along a roughly linear line:

When I do a linear regression of these values, I get a linear equation:
$$y = 0.997x-0.0136$$
In an ideal world, the equation should be $y = x$.
Clearly, my linear values are close to that ideal, but not exactly. My question is, how can I determine whether this result is statistically significant?
Is the value of 0.997 significantly different from 1? Is -0.01 significantly different from 0? Or are they statistically the same and I can conclude that $y=x$ with some reasonable confidence level?
What is a good statistical test I can use?
Thanks
 A: This type of situation can be handled by a standard F-test for nested models.  Since you want to test both of the parameters against a null model with fixed parameters, your hypotheses are:
$$H_0: \boldsymbol{\beta} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \quad \quad \quad H_A: \boldsymbol{\beta} \neq \begin{bmatrix} 0 \\ 1 \end{bmatrix} .$$
The F-test involves fitting both models and comparing their residual sum-of-squares, which are:
$$SSE_0 = \sum_{i=1}^n (y_i-x_i)^2 \quad \quad \quad SSE_A = \sum_{i=1}^n (y_i - \hat{\beta}_0 - \hat{\beta}_1 x_i)^2$$
The test statistic is:
$$F \equiv F(\mathbf{y}, \mathbf{x}) = \frac{n-2}{2} \cdot \frac{SSE_0 - SSE_A}{SSE_A}.$$
The corresponding p-value is:
$$p \equiv p(\mathbf{y}, \mathbf{x}) = \int \limits_{F(\mathbf{y}, \mathbf{x}) }^\infty \text{F-Dist}(r | 2, n-2) \ dr.$$

Implementation in R: Suppose your data is in a data-frame called DATA with variables called y and x.  The F-test can be performed manually with the following code.  In the simulated mock data I have used, you can see that the estimated coefficients are close to the ones in the null hypothesis, and the p-value of the test shows no significant evidence to falsify the null hypothesis that the true regression function is the identity function.
#Generate mock data (you can substitute your data if you prefer)
set.seed(12345);
n    <- 1000;
x    <- rnorm(n, mean = 0, sd = 5);
e    <- rnorm(n, mean = 0, sd = 2/sqrt(1+abs(x)));
y    <- x + e;
DATA <- data.frame(y = y, x = x);

#Fit initial regression model
MODEL <- lm(y ~ x, data = DATA);

#Calculate test statistic
SSE0   <- sum((DATA$y-DATA$x)^2);
SSEA   <- sum(MODEL$residuals^2);
F_STAT <- ((n-2)/2)*((SSE0 - SSEA)/SSEA);
P_VAL  <- pf(q = F_STAT, df1 = 2, df2 = n-2, lower.tail = FALSE);

#Plot the data and show test outcome
plot(DATA$x, DATA$y,
     main = 'All Residuals',
     sub  = paste0('(Test against identity function - F-Stat = ',
            sprintf("%.4f", F_STAT), ', p-value = ', sprintf("%.4f", P_VAL), ')'),
     xlab = 'Dataset #1 Normalized residuals',
     ylab = 'Dataset #2 Normalized residuals');
abline(lm(y ~ x, DATA), col = 'red', lty = 2, lwd = 2);

The summary output and plot for this data look like this:
summary(MODEL);

Call:
lm(formula = y ~ x, data = DATA)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.8276 -0.6742  0.0043  0.6703  5.1462 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -0.02784    0.03552  -0.784    0.433    
x            1.00507    0.00711 141.370   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.122 on 998 degrees of freedom
Multiple R-squared:  0.9524,    Adjusted R-squared:  0.9524 
F-statistic: 1.999e+04 on 1 and 998 DF,  p-value: < 2.2e-16

F_STAT;
[1] 0.5370824

P_VAL;
[1] 0.5846198


A: You could compute the coefficients with n bootstrapped samples. This will likely result in normal distributed coefficient values (Central limit theorem). With that you could then construct a (e.g. 95%) confidence interval with t-values (n-1 degrees of freedom) around the mean. If your CI does not include 1 (0), it is statistically significant different, or more precise: You can reject the null hypothesis of an equal slope.  
A: You could perform a simple test of hypothesis, namely a t-test. For the intercept your null hypothesis is $\beta_0=0$ (note that this is the significance test), and for the slope you have that under H0  $\beta_1=1$.
A: You should fit a linear regression and check the 95% confidence intervals for the two parameters. If the CI of the slope includes 1 and the CI of the offset includes 0 the two sided test is insignificant approx. on the (95%)^2 level -- as we use two separate tests the typ-I risk increases.
Using R:
fit = lm(Y ~ X)
confint(fit)

or you use
summary(fit)

and calc the 2 sigma intervals by yourself.
